Consecutive primes which are widely digitally delicate
aa r X i v : . [ m a t h . N T ] J a n Consecutive primeswhich are widely digitally delicate
Michael Filaseta
Dept. Mathematics, University of South Carolina, Columbia, SC 29208, USA [email protected]
Jacob Juillerat
Dept. Mathematics, University of South Carolina, Columbia, SC 29208, USA [email protected]
Dedicated to the fond memory of Ronald Graham
Abstract
We show that for every positive integer k , there exist k consecutive primes havingthe property that if any digit of any one of the primes, including any of the infinitelymany leading zero digits, is changed, then that prime becomes composite.
1. Introduction
In 1978, M. S. Klamkin [18] posed the following problem.
Does there exist any prime number such that if any digit (in base ) is changedto any other digit, the resulting number is always composite? In addition to computations establishing the existence of such a prime, the pub-lished solutions in 1979 to this problem included a proof by P. Erd˝os [6] that thereexist infinitely many such primes. Borrowing the terminology from J. Hopper andP. Pollack [15], we call such primes digitally delicate . The first digitally delicateprime is 294001. Thus, 294001 is a prime and, for every d ∈ { , , . . . , } , each ofthe numbers d , d , d , d , d , d is either equal to 294001 or composite. The proof provided by Erd˝os consistedof creating a partial covering system of the integers (defined in the next section)followed by a sieve argument. In 2011, T. Tao [28] showed by refining the sieveargument of Erd˝os that a positive proportion (in terms of asymptotic density) ofthe primes are digitally delicate. In 2013, S. Konyagin [20] pointed out that a similarapproach implies that a positive proportion of composite numbers n , coprime to 10,satisfy the property that if any digit in the base 10 representation of n is changed,then the resulting number remains composite. For example, the number n = 212159satisfies this property. Thus, every number in the set { d , d , d , d , d , d : d ∈ { , , , . . . , }} is composite. Later, in 2016, J. Hopper and P. Pollack [15] resolved a question ofTao’s on digitally delicate primes allowing for an arbitrary but fixed number of digitchanges to the beginning and end of the prime. All of these results and their proofshold for numbers written in an arbitrary base b rather than base 10, though theproof provided by Erd˝os [6] only addresses the argument in base 10.In 2020, the first author and J. Southwick [11] showed that a positive proportionof primes p , are widely digitally delicate , which they define as having the propertythat if any digit of p , including any one of the infinitely many leading zeros of p , isreplaced by any other digit, then the resulting number is composite. The proof wasspecific to base 10, though they elaborate on other bases for which the analogousargument produces a similar result, including for example base 31; however, it isnot even clear whether widely digitally delicate primes exist in every base. Observethat the first digitally delicate prime, 294001, is not widely digitally delicate since10294001 is prime. It is of some interest to note that even though a positive pro-portion of the primes are widely digitally delicate, no specific examples of widelydigitally delicate primes are known. Later in 2020, the authors with J. Southwick[9] gave a related argument showing that there are infinitely many (not necessarily apositive proportion) of composite numbers n in base 10 such that when any digit isinserted in the decimal expansion of n , including between two of the infinitely manyleading zeros of n and to the right of the units digit of n , the number n remainscomposite (see also [10]).In this paper, we show the following. Theorem 1.
For every positive integer k , there exist k consecutive primes all ofwhich are widely digitally delicate. Let P be a set of primes. It is not difficult to see that if P has an asymptoticdensity of 1 in the set of primes, then there exist k consecutive primes in P for each k ∈ Z + . On the other hand, for every ε ∈ (0 , P having asymptoticdensity 1 − ε in the set of primes such that there do not exist k consecutive primes in P for k sufficiently large (more precisely, for k ≥ /ε ). Thus, the prior results statedabove are not sufficient to establish Theorem 1. The main difficulty in using theprior methods to obtain Theorem 1 is in the application of sieve techniques in theprior work. We want to bypass the use of sieve techniques and instead give completecovering systems to show that there is an arithmetic progression containing infinitelymany primes such that every prime in the arithmetic progression is a widely digitallydelicate prime. This then gives an alternative proof of the result in [11]. After that,the main driving force behind the proof of Theorem 1, work of D. Shiu [25], canbe applied. D. Shiu [25] showed that in any arithmetic progression containinginfinitely many primes (that is, an + b with gcd( a, b ) = 1 and a >
0) there arearbitrarily long sequences of consecutive primes. Thus, once we establish throughcovering systems that such an arithmetic progression exists where every prime inthe arithmetic progression is widely digitally delicate, D. Shiu’s result immediatelyapplies to finish the proof of Theorem 1.Our main focus in this paper is on the proof of Theorem 1. However, in part,this paper is to emphasize that the remarkable work of Shiu [25] provides for a niceapplication to a number of results established via covering systems. One can alsotake these applications further by looking at the strengthening of Shiu’s work byJ. Maynard [22]. To illustrate the application of Shiu’s work in other context, wegive some further examples before closing this introduction.A Riesel number is a positive odd integer k with the property that k · n − n . A Sierpi´nski number is a positive oddinteger k with the property that k · n + 1 is composite for all nonnegative integers n . The existence of such k were established in [24] and [26], respectively, thoughthe former is a rather direct consequence of P. Erd˝os’s work in [5] and the latteris a somewhat less direct application of this same work, an observation made byA. Schinzel (cf. [8]). A Brier number is a number k which is simultaneously Rieseland Sierpi´nski, named after Eric Brier who first considered them (cf. [8]). Thesmallest known Brier number, discovered by Christophe Clavier in 2014 (see [27])is 3316923598096294713661 . As is common with all these numbers, examples typically come from covering sys-tems giving an arithmetic progression of examples. In particular, Clavier establishedthat every number in the arithmetic progression3770214739596601257962594704110 n + 3316923598096294713661 , n ∈ Z + ∪ { } is a Brier number. Since the numbers 3770214739596601257962594704110 and3316923598096294713661 are coprime, Shiu’s theorem gives the following. Theorem 2.
For every positive integer k , there exist k consecutive primes all ofwhich are Brier numbers. Observe that as an immediate consequence the same result holds if Brier numbersare replaced by Riesel or Sierpi´nski numbers.As another less obvious result to apply Shiu’s theorem to, we recall a result ofR. Graham [12] from 1964. He showed that there exist relatively prime positiveintegers a and b such that the recursive Fibonacci-like sequence u = a, u = b, and u n +1 = u n + u n − for integers n ≥ , (1)consists entirely of composite numbers. The known values for admissible a and b have decreased over the years through the work of others including D. Knuth [19],J. W. Nicol [23] and M. Vsemirnov [29], the latter giving the smallest known such a and b (but notably the same number of digits as the a and b in [23]). The result hasalso been generalized to other recursions; see A Dubickas, A. Novikas and J. ˇSiurys[4], D. Ismailescu, A. Ko, C. Lee and J. Y. Park [16] and I. Lunev [21]. As theGraham result concludes with all u i being composite, the initial elements of thesequence, a and b , are composite. However, there is still a sense in which one canapply Shiu’s result. To be precise, the smallest known example given by Vsemirnovis done by taking a = 106276436867 and b = 35256392432 . With u j defined as above, one can check that each u j is divisible by a prime fromthe set P = { , , , , , , , , , , , , , , , , } . Setting N = Y p ∈P p = 1821895895860356790898731230 , the value of a and b can be replaced by any integers a and b satisfying a ≡ N ) and b ≡ N ) . As gcd(106276436867 , N ) = 31 and gcd(35256392432 , N ) = 2, these congruencesare equivalent to taking a = 31 a ′ and b = 2 b ′ where a ′ and b ′ are integers satisfying a ′ ≡ b ′ ≡ . As a direct application of D. Shiu’s result, we have the following.
Theorem 3.
For every k ∈ Z + , there are k consecutive primes p , p , . . . , p k and k consecutive primes q , q , . . . , q k such that for any i ∈ { , , . . . , k } , the numbers a = 31 p i and b = 2 q i satisfy gcd( a, b ) = 1 and have the property that the u n definedby (1) are all composite. This latter result is not meant to be particularly significant but rather an indicationthat Shiu’s work does provide information in cases where covering systems are usedto form composite numbers.Regarding open problems, given the recent excellent works surrounding the non-existence of covering systems of particular forms (cf. [1, 2, 13, 14]), the authorsare not convinced that widely digitally delicate primes exist in every base. Thus,a tantalizing question is whether they exist or whether a positive proportion ofthe primes in every base are widely digitally delicate. In the opposite direction,as noted in [11], Carl Pomerance has asked for an unconditional proof that thereexist infinitely many primes which are not digitally delicate or which are not widelydigitally delicate. For other open problems in this direction, see the end of theintrodutcions in [9] and [11].
2. The first steps of the argument
As noted in the introduction, to prove Theorem 1, the work of D. Shiu [25] impliesthat it suffices to obtain an arithmetic progression An + B , with A and B relativelyprime positive integers, such that every prime in the arithmetic progression is widelydigitally delicate. We will determine such an A and B by finding relatively primepositive integers A and B satisfying property ( ∗ ) given by( ∗ ) If d ∈ {− , − , . . . , − } ∪ { , , . . . , } , then each number in the set A d = (cid:8) An + B + d · k : n ∈ Z + , k ∈ Z + ∪ { } (cid:9) is composite.As changing a digit of An + B , including any one of its infinitely many leading zerodigits, corresponds to adding or subtracting one of the numbers 1 , , . . . , An + B , we see that relatively prime positive integers A and B satisfyingproperty ( ∗ ) also satisfy the property we want, that every prime in An + B is widelydigitally delicate.To find relatively prime positive integers A and B satisfying property ( ∗ ), wemake use of covering systems which we define as follows. Definition 1.
A covering system (or covering) is a finite set of congruences x ≡ a (mod m ) , x ≡ a (mod m ) , . . . , x ≡ a r (mod m r ) , where r ∈ Z + , each a j ∈ Z , and each m j ∈ Z + , such that every integer satisfies atleast one congruence in the set of congruences. In other contexts in the literature, further restrictions can be made on the m j , sowe emphasize here that we want to allow for m j = 1 and for repeated moduli (sothat the m j are not necessarily distinct). There will be restrictions on the m j thatwill arise in the covering systems we build due to the approach we are using. Wewill see these as we proceed.For each d ∈ {− , − , . . . , − } ∪ { , , . . . , } , we will create a separate coveringsystem to show that the elements of A d in ( ∗ ) are composite. Table 1 indicates, foreach d , the number of different congruences in the covering system correspondingto d . Table 1: Number of congruences for each covering d − − − − − − d − − − d d are the exponents k on 10 in the definitionof A d . In other words, we will want to view each exponent k as satisfying one ofthe congruences in our covering system for a given A d . In the end, the values of A and B will be determined by the congruences we choose for the covering systemsas well as certain primes that arise in our method.We clarify that the work on digitally delicate primes in prior work mentioned inthe introduction used a partial covering of the integers k , that is a set of congruenceswhere most but not all integers k satisfy at least one of the congruences, togetherwith a sieve argument. The work in [11] on widely digitally delicate primes usedcovering systems for d ∈ { , , . . . , } and the same approach of partial coveringsand sieves for d ∈ {− , − , . . . , − } . The work in [9], like we will use in this paper,made use of covering systems for all d ∈ {− , − , . . . , − } ∪ { , , . . . , } . For [9],some of the covering systems could be handled rather easily by taking advantage ofthe fact that we were looking for composite numbers satisfying a certain propertyrather than primes.Next, we explain more precisely how we create and take advantage of a coveringsystem for a given fixed d ∈ {− , − , . . . , − } ∪ { , , . . . , } . We begin with acouple illustrative examples. Table 1 indicates that a number of the d are handledwith just one congruence. This is accomplished by taking A ≡ B ≡ . Observe that each element of A d in ( ∗ ) is divisible by 3 whenever d ≡ A and B are positive, as long as we also have B >
3, the elementsof A d for such d are all composite, which is our goal. Note the crucial role of theorder of 10 modulo the prime 3. The order is 1, and the covering system for eachof these d is simply k ≡ d because the choices for A and B , and hence the congruences on A and B above, areto be independent of d . For example, if d = 4, then An + B + d · k ≡ ≡ An + B + d · k will not be divisible by 3.As a second illustration, we see from Table 1 that we handle the digit d = 9 with4 congruences. The congruences for d = 9 are k ≡ , k ≡ , k ≡ , k ≡ . One easily checks that this is a covering system, that is that every integer k satisfiesone of these congruences. To take advantage of this covering system, we choosea different prime p for each congruence with 10 having order modulo p equal tothe modulus. We used the prime 11 with 10 of order 2, the prime 101 with 10of order 4, the prime 73 with 10 of order 8, and the prime 137 with 10 of order8. We take A divisible by each of these primes. For ( ∗ ), with d = 9, we want An + B + 9 · k composite. For k ≡ B ≡ B >
11 since then An + B + 9 · k ≡ B + 9 ≡ k ≡ B ≡
90 (mod 101) and
B >
101 since then An + B + 9 · k ≡
90 + 9 · ≡ ≡ k ≡ B ≡
56 (mod 73), we obtain An + B + 9 · k ≡ k ≡ B ≡
90 (mod 137), we obtain An + B + 9 · k ≡ B > ∗ ) holds with d = 9.Of some significance to our explanations later, we note that we could have in-terchanged the roles of the primes 73 and 137 since 10 has the same order for eachof these primes. In other words, we could associate 137 with the congruence k ≡ k ≡ k ≡ B ≡
47 (mod 137), we would have An + B + 9 · k ≡ k ≡ B ≡
17 (mod 73), we would have An + B + 9 · k ≡ k ≡ a (mod m ) in a covering system associated with a prime p for which the order of 10 modulo p is m , but how we choose the ordering of thoseprimes (which prime goes to which congruence) for a fixed modulus m is irrelevant.For each d ∈ {− , − , . . . , − } ∪ { , , . . . , } , we determine a covering system ofcongruences for k , where each modulus m corresponds to the order of 10 modulosome prime p . This imposes a condition on A , namely that A is divisible by each ofthese primes p . Fixing d , a congruence from our covering system k ≡ a (mod m ),and a corresponding prime p with 10 having order m modulo p , we determine B such that An + B + d · k ≡ B + d · a ≡ p ). Note that the values of d , a and p dictate the congruence condition for B modulo p . Each prime p willcorrespond to a unique congruence condition B ≡ − d · a (mod p ), so the ChineseRemainder Theorem implies the existence of a B ∈ Z + simultaneously satisfyingall the congruence conditions modulo primes on B . As long as B is large enough,then the condition ( ∗ ) will hold.To make sure that there is a prime of the form An + B , we will want gcd( A, B ) = 1.For k ≡ a (mod m ) and a corresponding prime p as above, we will have A divisibleby p and B ≡ − d · a (mod p ). Since d ∈ {− , − , . . . , − }∪{ , , . . . , } , if p ≥ p ∤ B . We will not be using the primes p ∈ { , } as 10 does nothave an order modulo these primes. We have already seen that we are using theprime p = 3 for d ≡ ∤ B . We will use p = 7 for d ∈ {− , − , − , − , − , , } , which then implies 7 ∤ B . Therefore, the conditiongcd( A, B ) = 1 will hold.Recall that we used the same congruence and corresponding prime in our coveringsystem for each d ≡ d if the corresponding prime, having 10 of order the modulus, is different.But in the case of d ≡ d . Toillustrate how we can repeat the use of a prime, we return to how we used theprime p = 11 above for d = 9. We ended up with A ≡ B ≡ p = 11 for d , we thereforewant An + B + d · k ≡ d · k ≡ d, k ) ∈ { ( − , , ( − , , (2 , , (9 , } . The case ( d, k ) = (9 ,
0) is from ourexample with d = 9 above. The case ( d, k ) = (2 ,
1) does not serve a purpose forus as d = 2 was covered by our earlier example using the prime 3 for all d ≡ d, k ) ∈ { ( − , , ( − , } are significant, and we makeuse of congruences modulo 11 in the covering systems for d = − d = −
2. Thus,we are able to repeat the use of some primes for different values of d . However, thisis not the case for most primes we used. A complete list of the primes which wewere able to use for more than one value of d is given in Table 2, together with thelist of corresponding d ’s. The function ρ ( m, p ) in this table will be explained in thenext section.Recalling that the modulus in a covering system is equal to the order of 10modulo a prime p , the role of primes and the order of 10 modulo those primesis significant in coming up with covering systems to deduce ( ∗ ). A modulus m can be used in a given covering system as many times as there are primes with10 of order m . Thus, for the covering system for d = 9, we saw the modulus 8being used twice as there are two primes with 10 of order 8, namely the primes 73and 137. One can look at a list of primitive prime factors of 10 k − m isthe same as the list of primes dividing Φ m (10) and not dividing m where Φ m ( x ) isthe m -th cyclotomic polynomial (cf. [3, 9, 11]). We used Magma V2.23-1 on a 2017MacBook Pro to determine different primes dividing Φ m (10). We did not always geta complete factorization but used that if the remaining unfactored part of Φ m (10) iscomposite, relatively prime to the factored part of Φ m (10) and m , and not a primepower, then there must be at least two further distinct prime factors of Φ m (10).This allowed us then to determine a lower bound on the number of distinct primesTable 2: Primes used for more than one digit d prime d ’s ρ ( m, p )3 − , − , − , , , − , − , − , − , − , , − , − , − , − , , − , − , − , − , − , − , − , − , − , , − , − , − , , − , − , , − , − , − , − , − , , − , − , − − , − , − , − , − d ’s ρ ( m, p )199 − , − , − , − , − , − , − , − − , − , − , − , − , − , − , − , − , − , m . Though we used most of these in our coverings, sometimes wefound extra primes that we did not need to use.In total, we made use of 673 different moduli m and 2596 different primes dividingΦ m (10) for such m . Of the 2596 different primes, there are 590 which came from 295composite numbers arising from an unfactored part of some Φ m (10), and there are63 other composite numbers for which only one prime factor of each of the compositenumbers was used. The largest explicit prime (not coming from the 295 + 63 = 358composite numbers) has 1700 digits, arising from testing what was initially a largeunfactored part of Φ m (10) for primality and determining it is a prime. The largestof the 358 composite numbers has 17234 digits. For obvious reasons, we will avoidlisting these primes and composites in this paper, though to help with verificationof the results, we are providing the data from our computations in [7]; more explicittables can also be found in [17].Table 4 in the appendix gives, for each of the 673 different moduli m , the detailedinformation on the number of distinct primes we used with 10 of order m , whichwe denote by L ( m ). Thus, L ( m ) is a lower bound on the total number of distinctprimes with 10 of order m . Note that L ( m ) is less than or equal to the number ofdistinct primes dividing Φ m (10) but not dividing m .For each d ∈ {− , − , . . . , − }∪{ , , . . . , } , the goal is to find a covering systemso that ( ∗ ) holds. We have already given the covering systems we obtained for d ≡ d = 9. In the next section and the appendix, we elaborate on thecovering systems for the remaining d . We also explain how the reader can verifythe data showing these covering systems satisfy the conditions needed for ( ∗ ).
3. Finishing the argument
To finish the argument, we need to present a covering system for each value of d in {− , − , . . . , − } ∪ { , , . . . , } as described in the previous section. For thepurposes of keeping the presentation of these covering systems manageable, for each m listed in Table 4, we take the L ( m ) primes we found with 10 of order m and orderthem in some way. Corresponding to the discussion concerning d = 9 and the primes73 and 137, the particular ordering is not important to us (for example, increasingorder would be fine). Suppose the primes corresponding to m are ordered in someway as p , p , . . . , p L ( m ) . We define ρ ( p j , m ) = j . Thus, if p j is the j -th prime inour ordering of the primes with 10 of order m , we have ρ ( p j , m ) = j . The particularvalues we used for ρ ( p j , m ) is not important to the arguments. So as to make theentries in Table 2 correct, the entries for ρ ( p, m ) indicate the values we used forthose primes. For example, Table 4 indicates there are 2 primes of order 6. One ofthem is 7. Table 2 indicates then that ρ (7 ,
6) = 1. Thus, we put 7 as the first ofthe 2 primes with 10 of order 6. The other prime with 10 of order 6 is 13, and asTable 2 indicates we set 13 as the second of the 2 primes with 10 of order 6.Tables 5-16 give the covering systems used for each d ∈ {− , − , . . . , − } ∪{ , , . . . , } with d k ≡ a (mod m )listed, we simply wrote the value of ρ ( m, p ). As m corresponds to the modulusused in the given congruence k ≡ a (mod m ) and the ordering of the primes is notsignificant to our arguments (any ordering will do), this is enough information toconfirm the covering arguments.That said, the time consuming task of coming up with the L ( m ) primes to orderfor each m is nontrivial (at least at this point in time). So that this work does notneed to be repeated, a complete list of the L ( m ) primes for each m is given in [7].Further, the tables in the form of lists can be found there as well, with the thirdcolumn in each case replaced by the prime we used with 10 of order the modulus ofthe congruence in the second column. In the way of clarity, recall that the primeswere not explicitly computed in the case that the unfactored part of Φ m (10) wastested to be composite; instead the composite number is listed in place of bothprimes in [7].For the remainder of this section, we clarify how to verify the information in Ta-bles 5-16. We address both verification of the covering systems and the informationon the primes as listed in [7].1 The most direct way to check that a system C of congruences x ≡ a (mod m ) , x ≡ a (mod m ) , . . . , x ≡ a s (mod m s )is a covering system is to set ℓ = lcm( m , m , . . . , m s ) and then to check if everyinteger in the interval [0 , ℓ −
1] satisfies at least one congruence in C . If not, then C is not a covering system. If on the other hand, every integer in [0 , ℓ −
1] satisfies acongruence in C , then C is a covering system. To see the latter, let n be an arbitraryinteger, and write n = ℓq + r where q and r are integers with 0 ≤ r ≤ ℓ −
1. Since r ∈ [0 , ℓ −
1] satisfies some x ≡ a j (mod m j ) and since ℓ ≡ m j ), we deducefor this same j that n = ℓq + r ≡ a j (mod m j ).The above is a satisfactory approach if ℓ is not too large. For the values of d in {− , − , . . . , − } ∪ { , , . . . , } with d ℓ given by the congruences in Tables 5-16 are listed in Table 3. The maximumprime divisor of ℓ is also listed in the fourth column of Table 3. The value of ℓ canexceed 10 , so we found a more efficient way to test whether one of our systems C of congruences, where ℓ is large, is a covering system.Table 3: Least common multiple of the moduli for the coverings in each table d Table ℓ max p − − − − − − d Table ℓ max p ℓ > in Table 3 and the corresponding collection of congruencescoming from the table indicated in the second column is C . Let q be the largestprime divisor of ℓ as indicated in the fourth column. Let w = 4 · · · q . This choiceof w was selected on the basis of some trial and error; other choices are certainlyreasonable. We do however want and have that w divides ℓ . Based on the commentsabove, we would like to know if every integer in the interval [0 , ℓ −
1] satisfies atleast one congruence in C . The basic idea is to take each u ∈ [0 , w −
1] and toconsider the integers that are congruent to u modulo w in [0 , ℓ − C needs to be considered. For example,take d = −
3. Then Table 3 indicates ℓ = 1486147703040 and Table 1 indicates thenumber of congruences in C is 739. From Table 9, the first few of the congruencesin C are k ≡ , k ≡ , k ≡ , k ≡
11 (mod 21) . w = 4 · · ·
19 = 1140. If we take u = 0, then only the third of thesecongruences can be satisfied by an integer k congruent to u modulo w , as eachof the other ones requires k k ≡ u (mod w ) requires k ≡ C ′ be the congruences in C which are consistent with k ≡ u (mod w ).One can determine these congruences by using that there exist integers satisfyingboth k ≡ a (mod m ) and k ≡ u (mod w ) if and only if a ≡ u (mod gcd( m, w )).Observe that, with u ∈ [0 , w −
1] fixed, we would like to know if each integer v of the form v = wt + u, with 0 ≤ t ≤ ( ℓ/w ) − C ′ . The main advantage of this approach is that,as we shall now see, not all ℓ/w values of t need to be considered. First, we notethat if C ′ is the empty set, then the integers in (2) are not covered and therefore C is not a covering system. Suppose then that |C ′ | ≥
1. Let ℓ ′ denote the leastcommon multiple of the moduli in C ′ . Let δ = gcd( w, ℓ ′ ). We claim that we needonly consider v = wt + u where 0 ≤ t ≤ ( ℓ ′ /δ ) −
1. To see this, suppose we knowthat every v = wt + u with 0 ≤ t ≤ ( ℓ ′ /δ ) − C ′ .There are integers q , q ′ , r and r ′ satisfying t = ℓ ′ q ′ + r ′ where 0 ≤ r ′ ≤ ℓ ′ − r ′ = ( ℓ ′ /δ ) q + r , where 0 ≤ r ≤ ( ℓ ′ /δ ) −
1. Then v = wt + u = wℓ ′ q ′ + wr ′ + u = wℓ ′ q ′ + ( w/δ ) ℓ ′ q + wr + u. The definition of δ implies that w/δ ∈ Z . As each modulus in C ′ divides ℓ ′ , wesee that v satisfies a congruence in C ′ if and only if wr + u does. Here, w and u are fixed and 0 ≤ r ≤ ( ℓ ′ /δ ) −
1. Thus, we see that for each u ∈ [0 , w − v in (2) satisfies a congruence in C ′ for0 ≤ t ≤ ( ℓ ′ /δ ) −
1. Returning to the example of d = − ℓ = 1486147703040and |C| = 739, where w = 1140 and we considered u = 0, one can check that |C ′ | = 19, ℓ ′ = 12640320, δ = w and ℓ ′ /δ = 11088. Thus, what started outas ominously checking whether over 10 integers each satisfy at least one of 739different congruences is reduced in the case of u = 0 to looking at whether 11088integers each satisfy at least one of 19 different congruences. As u ∈ [0 , w −
1] varies,the number of computations does as well. An extreme case for d = − u = 75, where we get ℓ ′ /δ = 14325696 and |C ′ | = 47. As d and u vary, though, thiscomputation becomes manageable for determining that we have covering systemsfor each d in {− , − , . . . , − } ∪ { , , . . . , } with d ℓ > .On a 2017 MacBook Plus running Maple 2019 with a 2.3 GHz Dual-Core IntelCore i5 processor, the total cpu time for determining the systems of congruences inTables 5-16 are all covering systems took approximately 2 . d = − . ℓ ′ /δ encountered was ℓ ′ /δ = 14325696 which occurred precisely for d = − u ∈ { , , , , } .3 The most cumbersome task for us was the determination of the data in Table 4. Asnoted earlier, although the reader can check the data there directly, we have madethe list of primes corresponding to each m available through [7]. With the list ofsuch primes for each m , it is still worth indicating how the data can be checked.Recall, in particular, the list of primes is not explicit in the case that there wasan unfactored part of Φ m (10). In this subsection, we elaborate on what checksshould be and were done. All computations below were done with the MacBookPro mentioned at the end of the last subsection and using Magma V2.23-1.For each modulus m used in our constructions (listed in Table 4), we made alist of primes p , p , . . . , p s , written in increasing order, together with up to twoadditional primes q and q , included after p s on the list but not written explicitly(as we will discuss). Each prime came from a factorization or partial factorization ofΦ m (10). The primes p , p , . . . , p s are the distinct primes appearing in the factoredpart of Φ m (10), and as noted earlier do not include primes dividing m . In somecases, a complete factorization was found for Φ m (10). For such m , there are noadditional primes q and q . If Φ m (10) had an unfactored part Q > Q is relatively prime to mp p · · · p s and that Q is not of the form N k with N ∈ Z + and k an integer greater than orequal to 2. As this was always the case for the Q tested, we knew each such Q had two distinct prime factors q and q . We deduce that there are at least twomore primes q j , j ∈ { , } , different from p , p , . . . , p s for which 10 has order m modulo q j . As the data only contains the primes used in the covering systems, weonly included the primes q and q that were used. Thus, despite Q having at leasttwo distinct prime divisors, we may have listed anywhere from 0 to 2 of them. Thequestion arises, however, as to how one can list primes that we do not know; thereare primes q and q dividing Q , but we were unable to (or chose not to) factor Q to determine them explicitly. Instead of listing q and q then, we opted to list Q .Thus, for each m we associated a list of one of the forms[ p , p , . . . , p s ] , [ p , p , . . . , p s , Q ] , [ p , p , . . . , p s , Q, Q ] , depending on whether Q either did not exist or we used no prime factor of Q , weused one prime factor of Q , or we used two prime factors of Q , respectively. It ispossible that s = 0; for example, the lists associated with the moduli 2888 and 2976each take the middle form with no p j and one composite number.For a fixed m , given such a list, say from [7], one merely needs to check: • Each element of the list divides Φ m (10). • Each element of the list is relatively prime to m . • There is at most one composite number, say
Q >
1, in the list, which mayappear at most twice. The other numbers in the list are distinct primes.4 • If the composite number Q exists, then gcd( Q, p p · · · p s ) = 1. • If the composite number Q exists twice, then Q /k Z + for every integer k ∈ [2 , log( Q ) / log(2)].The upper bond in the last item above is simply because k > log( Q ) / log(2) implies1 < Q /k < Q /k is not an integer. For each m , the value of L ( m ) inTable 4 is simply the number of elements in the list associated with m .With the data from the tables in the Appendix, also available in [7] with theindicated primes p , p , . . . , p s , q , q depending on m as above, some further detailsneed to be checked to fully justify the computations. We verified that whenever m is used as a modulus in a table, it was associated with one of the primes dividingΦ m (10). Furthermore, for any given d ∈ {− , − , . . . , − } ∪ { , , . . . , } , the com-plete list of primes used as the congruences vary are distinct, noting that q and q ,for a given m , will be denoted by the same number Q but represent two distinctprime divisors of Q . As d varies, a given modulus m and a prime p dividing Φ m (10)can be used more than once as indicated in Table 2. To elaborate, suppose suchan m and p is used for each d ∈ D ⊆ {− , − , . . . , − } ∪ { , , . . . , } . For each d ∈ D , then, there corresponds a congruence k ≡ a (mod m ), where a = a ( d ) willdepend on d , as well as m and p . As noted earlier, this is permissible if and onlyif the values of d · a ( d ) are congruent modulo p for all d ∈ D . Thus, for each p that occurs in more than one table, as in Table 2, a check is done to verify thecorresponding values of d · a ( d ) are congruent modulo p .The verification of the covering systems needed for Theorem 1 is complete, andthe work of D. Shiu [25] now implies the theorem. References [1] P. Balister, B. Bollob´as, R. Morris, J. Sahasrabudhe and M. Tiba,
The Erd˝os–Selfridge problem with square-free moduli , Algebra & Number Theory, to ap-pear.[2] P. Balister, B. Bollob´as, R. Morris, J. Sahasrabudhe and M. Tiba,
On the Erd˝osCovering Problem: the density of the uncovered set , arXiv:1811.03547.[3] J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff,Jr., Factorizations of b n ± b = 2 , , , , , , ,
12 Up to High Powers, 3rdedition, Contemporary Mathematics, Vol. 22, American Math. Soc., Provi-dence, 2002 (available online).[4] A Dubickas, A. Novikas and J. ˇSiurys,
A binary linear recurrence sequence ofcomposite numbers , J. Number Theory 130 (2010), 1737–1749.5[5] P. Erd˝os,
On integers of the form k + p and some related problems , SummaBrasil. Math. 2 (1950), 113–123.[6] P. Erd˝os, Solution to problem 1029: Erdos and the computer , MathematicsMagazine 52 (1979), 180-181.[7] M. Filaseta and J. Juillerat, Data for “Consecutive primes which arewidely digitally delicate” (containing data for Table 4 and Tables 5-16),https://people.math.sc.edu/filaseta/ConsecutiveWDDPrimes.html.[8] M. Filaseta, C. Finch and M. Kozek,
On powers associated with Sierpi´nskinumbers, Riesel numbers and Polignac’s conjecture , J. Number Theory 128(2008), 1916–1940.[9] M. Filaseta, J. Juillerat and J. Southwick,
Widely Digitally Stable Numbers ,Combinatorial and Additive Number Theory IV, Springer, to appear.[10] M. Filaseta, M. Kozek, C. Nicol and J. Selfridge,
Composites that remain com-posite after changing a digit , Journal of Combinatorics and Number Theory 2(2010), 25–36.[11] M. Filaseta and J. Southwick,
Primes that become composite after changing anarbitrary digit , Mathematics of Computation, to appear.[12] R. Graham,
A Fibonacci-like sequence of composite numbers , Math. Mag. 37(1964), 322–324.[13] Robert D. Hough,
Solution of the minimum modulus problem for covering sys-tems , Annals of Math 181 (2015), 361–382.[14] Robert D. Hough and Pace P. Nielsen,
Covering systems with restricted divis-ibility , Duke Math. J. 168 (2019), 3261–3295.[15] J. Hopper and P. Pollack,
Digitally delicate primes , J. Number Theory 168(2016), 247–256.[16] D. Ismailescu, A. Ko, C. Lee and J. Y. Park,
On second-order linear sequencesof composite numbers , J. Integer Seq. 22 (2019), Art. 19.7.2, 16 pp.[17] J. Juillerat, Widely digitally stable numbers and irreducibility criteria for poly-nomials with prime values (tentative title), University of South Carolina, dis-sertation, 2021 (expected).[18] M. S. Klamkin,
Problem 1029 , Mathematics Magazine 51 (1978), p. 69.[19] D. Knuth,
A Fibonacci-like sequence of composite numbers , Math. Mag. 63(1990), 21–25.6[20] S. V. Konyagin,
Numbers that become composite after changing one or twodigits , Pioneer Jour. of Algebra, Number Theory and Appl. 6 (2013), 1–7.[21] I. Lunev,
A tribonacci-like sequence of composite numbers , J. Integer Seq. 20(2017), Art. 17.3.2, 6 pp.[22] J. Maynard,
Dense clusters of primes in subsets , Compositio Math. 152 (2016),1517–1554.[23] J. W. Nicol,
A Fibonacci-like sequence of composite numbers , Electron. J. Com-bin. 6 (1999),
N˚agra stora primtal , Elementa 39 (1956), 258–260.[25] D. K. L. Shiu,
Strings of congruent primes , J. Lond. Math. Soc. 61 (2000),359–373.[26] W. Sierpi´nski,
Sur un probl`eme concernant les nombres k · n +1, Elem. Math. 15(1960), 73–74.[27] N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, pub-lished electronically at https://oeis.org/A076335, Oct. 29, 2020.[28] T. Tao, A remark on primality testing and decimal expansions , J. Aust. Math.Soc. 91 (2011), 405–413.[29] M. Vsemirnov,
A new Fibonacci-like sequence of composite numbers , J. IntegerSeq. 7 (2004), Article 04.3.7, 3 pp.
Appendix
This appendix begins with Table 4, which gives a lower bound L ( m ) on the numberof distinct prime divisors of Φ m (10) that are relatively prime to m . The m listedcorrespond to moduli used in our coverings. The number L ( m ) also provides a lowerbound on the number of primes p for which 10 has order m modulo p .After Table 4, the remaining Tables 5-16 give the congruences k ≡ a (mod m )that form the covering systems we obtained for d ∈ {− , − , . . . , − } ∪ { , , . . . , } with d ρ ( m, p ) discussed earlier in this paper).7Table 4: Number of primes used, L = L ( m ), with 10 of order m (Part I) m L m L
56 257 358 260 362 163 364 465 266 268 369 370 272 374 375 376 277 478 480 281 584 285 387 388 290 291 792 393 195 596 499 4100 4102 2104 2105 3108 3110 4111 3112 2114 2115 5116 4117 4119 4120 1121 4124 2125 4126 2130 2 m L
132 3133 3135 5136 2138 3140 5143 4144 2145 4148 6152 4153 6154 6155 3156 3161 5162 4165 3168 3169 3170 3171 3174 3175 3176 2180 3182 2184 2185 4186 4187 3190 3192 4195 3198 2203 4204 6207 3208 6209 5210 3216 2217 3220 6221 3222 4228 7230 2231 6232 5 m L
234 3238 3240 3242 5247 5248 5250 2252 2253 5255 4259 5260 5261 8264 7266 3270 4272 2273 5275 5276 5280 2285 3286 5289 3290 6296 3297 6299 3304 3306 6310 4312 5315 3319 4322 6323 3330 3333 4336 3338 4340 5341 5342 5345 6348 5350 2351 4352 4357 6360 3 m L
361 7363 3364 3368 2370 2372 3374 5377 4380 8384 5390 3391 4396 3399 5403 5406 5407 6408 5414 5416 5418 4420 4425 3429 3432 2434 5435 4437 5440 2442 5444 4455 5456 5459 5460 8462 3464 7465 5476 5480 3483 7484 6494 4495 6496 4506 3507 7510 5513 2518 2 m L
520 2522 4527 4528 3532 5540 7544 3546 2552 4555 6561 4570 4572 5575 6578 5580 4589 2592 2594 5595 4598 3605 2608 3609 3612 6620 2621 4624 2627 4630 5638 5644 3646 7651 3660 8663 3665 3666 4672 6676 3680 4682 6684 4690 5693 6696 3702 4704 4714 2715 3 L = L ( m ), with 10 of order m (Part II) m L
720 2722 3726 4728 2740 10741 5744 7748 4754 2759 3760 2765 2768 6777 5782 6792 2798 3805 4806 3812 6814 5816 4828 2833 5836 3840 5845 2850 1858 5867 8868 6870 7874 5880 5884 4888 4897 3910 5912 2918 4920 2924 5925 9928 4930 4931 7935 5952 6960 2966 3 m L
969 4988 3990 4992 41001 41012 31014 41015 71020 71023 41026 41035 31036 21040 41044 61045 31054 21056 31064 31083 71085 31088 61104 71105 41110 21122 21131 41140 111150 21156 51160 21173 41178 51183 41188 21190 61196 71197 31209 31210 31216 91221 61224 51235 51240 61242 41254 51260 71275 41276 3 m L m L m L m L L = L ( m ), with 10 of order m (Part III) m L m L m L m L m L m L Table 5: Covering information for d = − congruence pk ≡ k ≡ k ≡ k ≡ k ≡
26 (mod 28) 1 k ≡
16 (mod 22) 1 k ≡
26 (mod 33) 1 k ≡ k ≡ k ≡ k ≡ k ≡
34 (mod 62) 1 k ≡
35 (mod 93) 1 k ≡
98 (mod 186) 1 k ≡
68 (mod 186) 2 k ≡
38 (mod 186) 3 k ≡ k ≡
40 (mod 124) 1 k ≡
102 (mod 124) 2 k ≡
320 (mod 372) 1 k ≡
134 (mod 372) 2 k ≡
104 (mod 372) 3 k ≡
42 (mod 248) 1 k ≡
136 (mod 248) 2 k ≡
74 (mod 248) 3 k ≡
12 (mod 248) 4 k ≡
168 (mod 248) 5 k ≡
602 (mod 744) 1 congruence pk ≡
44 (mod 744) 2 k ≡
200 (mod 744) 4 k ≡
386 (mod 744) 3 k ≡
572 (mod 744) 5 k ≡
728 (mod 744) 6 k ≡
170 (mod 744) 7 k ≡
414 (mod 496) 1 k ≡
446 (mod 496) 2 k ≡
478 (mod 496) 3 k ≡
14 (mod 496) 4 k ≡
662 (mod 992) 1 k ≡
694 (mod 992) 2 k ≡
726 (mod 992) 3 k ≡
758 (mod 992) 4 k ≡
356 (mod 1488) 1 k ≡ k ≡
542 (mod 1488) 3 k ≡ k ≡
140 (mod 155) 1 k ≡
16 (mod 155) 2 k ≡
47 (mod 155) 3 k ≡
78 (mod 310) 1 k ≡
264 (mod 310) 2 k ≡
110 (mod 310) 3 k ≡
296 (mod 310) 4 k ≡
17 (mod 465) 1 k ≡
203 (mod 465) 2 k ≡
389 (mod 465) 3 congruence pk ≡
80 (mod 465) 4 k ≡
266 (mod 465) 5 k ≡
452 (mod 930) 1 k ≡
638 (mod 930) 2 k ≡
824 (mod 930) 3 k ≡
50 (mod 930) 4 k ≡
236 (mod 620) 1 k ≡
546 (mod 620) 2 k ≡
112 (mod 1240) 1 k ≡ k ≡
732 (mod 1240) 4 k ≡
422 (mod 1240) 3 k ≡
608 (mod 1240) 5 k ≡ k ≡ k ≡ k ≡
794 (mod 1860) 4 k ≡
20 (mod 1860) 3 k ≡
950 (mod 1860) 5 k ≡ k ≡
206 (mod 1860) 7 k ≡
392 (mod 1860) 8 k ≡ k ≡ k ≡
578 (mod 1395) 1 k ≡
113 (mod 1395) 2 k ≡ k ≡
299 (mod 1395) 4 d = − congruence pk ≡ k ≡
764 (mod 2790) 1 k ≡
21 (mod 217) 1 k ≡
176 (mod 217) 2 k ≡
114 (mod 217) 3 k ≡
52 (mod 434) 1 k ≡
424 (mod 434) 2 k ≡
300 (mod 434) 3 k ≡
84 (mod 434) 4 k ≡
22 (mod 434) 5 k ≡
611 (mod 651) 1 k ≡
332 (mod 651) 2 k ≡
53 (mod 651) 3 k ≡
146 (mod 1302) 1 k ≡ k ≡
302 (mod 1302) 3 k ≡
674 (mod 1302) 4 k ≡
796 (mod 868) 1 k ≡
208 (mod 868) 2 k ≡
488 (mod 868) 3 k ≡
768 (mod 868) 4 k ≡
612 (mod 868) 5 k ≡
178 (mod 868) 6 k ≡
116 (mod 2604) 1 k ≡ k ≡
860 (mod 2604) 3 k ≡ k ≡
644 (mod 2604) 5 k ≡
210 (mod 1736) 1 k ≡ k ≡
365 (mod 1085) 1 k ≡ k ≡
582 (mod 1085) 3 k ≡
148 (mod 2170) 1 k ≡ k ≡
520 (mod 2170) 3 k ≡
86 (mod 2170) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
458 (mod 6510) 2 k ≡ k ≡
830 (mod 1953) 1 k ≡ k ≡
179 (mod 1953) 3 k ≡
272 (mod 1953) 4 k ≡
923 (mod 1953) 5 congruence pk ≡ k ≡
242 (mod 341) 1 k ≡
56 (mod 341) 2 k ≡
211 (mod 341) 3 k ≡
25 (mod 341) 4 k ≡
149 (mod 341) 5 k ≡
304 (mod 682) 1 k ≡
118 (mod 682) 2 k ≡
614 (mod 682) 3 k ≡
428 (mod 682) 4 k ≡
88 (mod 682) 5 k ≡
584 (mod 682) 6 k ≡
398 (mod 1023) 1 k ≡
212 (mod 1023) 2 k ≡
677 (mod 1023) 3 k ≡
491 (mod 1023) 4 k ≡ k ≡ k ≡
956 (mod 2046) 3 k ≡
182 (mod 403) 1 k ≡
27 (mod 403) 2 k ≡
275 (mod 403) 3 k ≡
120 (mod 403) 4 k ≡
368 (mod 403) 5 k ≡
616 (mod 806) 1 k ≡
58 (mod 806) 2 k ≡
306 (mod 806) 3 k ≡
554 (mod 1209) 1 k ≡
647 (mod 1209) 2 k ≡
89 (mod 1209) 3 k ≡
740 (mod 2418) 1 k ≡
338 (mod 2418) 2 k ≡ k ≡ k ≡ k ≡
524 (mod 1612) 1 k ≡
772 (mod 1612) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
710 (mod 4836) 5 k ≡
803 (mod 3627) 1 k ≡ k ≡ k ≡
245 (mod 3627) 4 k ≡ pk ≡ k ≡ k ≡
896 (mod 14508) 1 k ≡ k ≡ k ≡ k ≡
10 (mod 34) 1 k ≡
153 (mod 527) 1 k ≡
494 (mod 527) 2 k ≡
308 (mod 527) 3 k ≡
122 (mod 527) 4 k ≡
990 (mod 1054) 1 k ≡
804 (mod 1054) 2 k ≡ k ≡
959 (mod 1581) 2 k ≡
773 (mod 1581) 3 k ≡
587 (mod 1581) 4 k ≡
215 (mod 1581) 5 k ≡
29 (mod 1581) 6 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
866 (mod 6324) 3 k ≡
247 (mod 589) 1 k ≡
495 (mod 589) 2 k ≡
154 (mod 1178) 1 k ≡
402 (mod 1178) 2 k ≡
650 (mod 1178) 3 k ≡
898 (mod 1178) 4 k ≡ k ≡ k ≡
464 (mod 1767) 2 k ≡ k ≡
371 (mod 1767) 4 k ≡ k ≡
278 (mod 3534) 2 k ≡ k ≡ k ≡ k ≡ k ≡
92 (mod 2356) 3 k ≡ k ≡ k ≡ k ≡ k ≡ d = − congruence pk ≡ k ≡ k ≡
19 (mod 28) 1 k ≡
10 (mod 22) 1 k ≡ k ≡ k ≡
11 (mod 15) 1 k ≡
13 (mod 16) 1 k ≡ k ≡ k ≡ k ≡
19 (mod 34) 2 k ≡ k ≡
88 (mod 204) 1 k ≡
20 (mod 204) 2 k ≡
190 (mod 204) 3 k ≡
122 (mod 204) 4 k ≡ k ≡
21 (mod 68) 2 k ≡
38 (mod 68) 3 k ≡
123 (mod 136) 1 k ≡
55 (mod 136) 2 k ≡ k ≡
243 (mod 272) 2 k ≡
651 (mod 1632) 4 k ≡ k ≡ k ≡ k ≡
107 (mod 3264) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
379 (mod 4896) 1 k ≡ k ≡ k ≡ k ≡
923 (mod 9792) 1 k ≡ k ≡ k ≡ k ≡
39 (mod 544) 1 k ≡
311 (mod 544) 2 k ≡
175 (mod 544) 3 k ≡ k ≡ k ≡ k ≡ k ≡ pk ≡ k ≡
481 (mod 2176) 2 k ≡
73 (mod 1088) 1 k ≡
209 (mod 1088) 2 k ≡
345 (mod 1088) 3 k ≡
617 (mod 1088) 4 k ≡
753 (mod 1088) 5 k ≡
889 (mod 1088) 6 k ≡
56 (mod 459) 1 k ≡
413 (mod 459) 2 k ≡
311 (mod 459) 3 k ≡
209 (mod 459) 4 k ≡
107 (mod 459) 5 k ≡
464 (mod 918) 1 k ≡
362 (mod 918) 2 k ≡
260 (mod 918) 3 k ≡
158 (mod 918) 4 k ≡
124 (mod 204) 5 k ≡
22 (mod 204) 6 k ≡
447 (mod 1632) 1 k ≡
991 (mod 1632) 2 k ≡ k ≡
40 (mod 51) 1 k ≡
23 (mod 51) 2 k ≡ k ≡
41 (mod 51) 4 k ≡
57 (mod 102) 1 k ≡
75 (mod 102) 2 k ≡
127 (mod 153) 1 k ≡
110 (mod 153) 2 k ≡
76 (mod 153) 3 k ≡
59 (mod 153) 4 k ≡
25 (mod 153) 5 k ≡ k ≡
297 (mod 306) 1 k ≡
93 (mod 306) 2 k ≡
195 (mod 306) 3 k ≡
145 (mod 408) 1 k ≡
43 (mod 408) 2 k ≡
349 (mod 408) 3 k ≡
247 (mod 408) 4 k ≡
128 (mod 408) 5 k ≡
434 (mod 816) 1 k ≡
26 (mod 816) 2 k ≡
740 (mod 816) 3 k ≡
332 (mod 816) 4 k ≡
230 (mod 2040) 1 k ≡ k ≡ k ≡
638 (mod 2040) 4 congruence pk ≡ k ≡
111 (mod 612) 1 k ≡ k ≡
519 (mod 612) 3 k ≡
417 (mod 612) 4 k ≡
315 (mod 612) 5 k ≡
213 (mod 612) 6 k ≡
94 (mod 306) 4 k ≡
298 (mod 306) 5 k ≡
196 (mod 306) 6 k ≡
77 (mod 1224) 1 k ≡ k ≡
893 (mod 1224) 3 k ≡
689 (mod 1224) 4 k ≡
485 (mod 1224) 5 k ≡ k ≡
281 (mod 2448) 2 k ≡
995 (mod 1020) 1 k ≡
791 (mod 1020) 2 k ≡
587 (mod 1020) 3 k ≡
383 (mod 1020) 4 k ≡
179 (mod 1020) 5 k ≡
10 (mod 85) 1 k ≡
61 (mod 85) 2 k ≡
27 (mod 85) 3 k ≡
78 (mod 170) 1 k ≡
163 (mod 170) 2 k ≡
44 (mod 170) 3 k ≡
129 (mod 680) 1 k ≡
299 (mod 680) 2 k ≡
469 (mod 680) 3 k ≡
639 (mod 680) 4 k ≡
215 (mod 255) 1 k ≡
62 (mod 255) 2 k ≡
113 (mod 255) 3 k ≡
164 (mod 255) 4 k ≡
45 (mod 510) 1 k ≡
351 (mod 510) 2 k ≡
147 (mod 510) 3 k ≡
453 (mod 510) 4 k ≡
249 (mod 510) 5 k ≡
640 (mod 1020) 6 k ≡
385 (mod 1020) 7 k ≡
130 (mod 4080) 2 k ≡ k ≡ k ≡ k ≡ k ≡
895 (mod 3060) 1 k ≡ d = − congruence pk ≡
181 (mod 765) 1 k ≡
436 (mod 765) 2 k ≡ k ≡
691 (mod 1530) 2 k ≡ k ≡
487 (mod 1530) 3 k ≡
742 (mod 1530) 5 k ≡ k ≡
232 (mod 3060) 4 k ≡
997 (mod 3060) 6 k ≡ k ≡ k ≡
28 (mod 425) 1 k ≡
283 (mod 425) 2 k ≡
113 (mod 425) 3 k ≡
368 (mod 1700) 1 k ≡
793 (mod 1700) 2 k ≡ k ≡ k ≡ k ≡
198 (mod 1700) 6 k ≡
623 (mod 850) 1 k ≡
79 (mod 1275) 1 k ≡
334 (mod 1275) 2 k ≡
589 (mod 1275) 3 k ≡
844 (mod 1275) 4 k ≡ k ≡ k ≡
63 (mod 119) 1 k ≡
29 (mod 119) 2 k ≡
114 (mod 119) 3 k ≡
80 (mod 119) 4 k ≡
46 (mod 238) 1 k ≡
165 (mod 238) 2 k ≡
12 (mod 238) 3 k ≡
131 (mod 476) 1 k ≡
216 (mod 476) 2 k ≡
97 (mod 476) 3 k ≡
454 (mod 476) 4 k ≡
335 (mod 476) 5 k ≡ k ≡ k ≡ k ≡
369 (mod 3808) 2 k ≡
336 (mod 952) 1 k ≡
217 (mod 952) 2 k ≡
98 (mod 952) 3 k ≡
931 (mod 952) 4 k ≡
812 (mod 952) 5 k ≡
574 (mod 952) 6 congruence pk ≡
693 (mod 1904) 3 k ≡
455 (mod 1904) 4 k ≡ k ≡
64 (mod 357) 1 k ≡
302 (mod 357) 2 k ≡
268 (mod 357) 3 k ≡
149 (mod 357) 4 k ≡
115 (mod 357) 5 k ≡
353 (mod 357) 6 k ≡
183 (mod 714) 1 k ≡
387 (mod 714) 2 k ≡ k ≡
591 (mod 1428) 2 k ≡
200 (mod 595) 1 k ≡
557 (mod 595) 2 k ≡
438 (mod 595) 3 k ≡
319 (mod 595) 4 k ≡
676 (mod 1785) 1 k ≡
81 (mod 1785) 2 k ≡
880 (mod 1190) 2 k ≡
761 (mod 1190) 1 k ≡
642 (mod 1190) 3 k ≡
523 (mod 1190) 4 k ≡
404 (mod 1190) 5 k ≡
285 (mod 1190) 6 k ≡
166 (mod 2380) 1 k ≡ k ≡
47 (mod 2380) 3 k ≡ k ≡ k ≡ k ≡ k ≡
999 (mod 3570) 4 k ≡ k ≡ k ≡
251 (mod 833) 1 k ≡
13 (mod 833) 2 k ≡
608 (mod 833) 3 k ≡
370 (mod 833) 4 k ≡
132 (mod 833) 5 k ≡ k ≡
727 (mod 1666) 2 k ≡ k ≡
489 (mod 1666) 4 k ≡
99 (mod 187) 1 k ≡
133 (mod 187) 2 k ≡
167 (mod 187) 3 k ≡
14 (mod 374) 1 k ≡
201 (mod 374) 2 k ≡
48 (mod 374) 3 congruence pk ≡
235 (mod 374) 4 k ≡
65 (mod 374) 5 k ≡
456 (mod 748) 1 k ≡
269 (mod 748) 2 k ≡
82 (mod 748) 3 k ≡
643 (mod 748) 4 k ≡
592 (mod 1496) 1 k ≡
218 (mod 1496) 2 k ≡ k ≡
966 (mod 1496) 4 k ≡
405 (mod 1870) 1 k ≡
31 (mod 1870) 2 k ≡ k ≡ k ≡ k ≡ k ≡
490 (mod 935) 1 k ≡
116 (mod 935) 2 k ≡
677 (mod 935) 3 k ≡
303 (mod 935) 4 k ≡
864 (mod 935) 5 k ≡
337 (mod 561) 1 k ≡
524 (mod 561) 2 k ≡
184 (mod 561) 3 k ≡
371 (mod 561) 4 k ≡
711 (mod 1122) 1 k ≡ k ≡
117 (mod 221) 1 k ≡
66 (mod 221) 2 k ≡
185 (mod 221) 3 k ≡
134 (mod 442) 1 k ≡
355 (mod 442) 2 k ≡
304 (mod 442) 3 k ≡
83 (mod 442) 4 k ≡
32 (mod 442) 5 k ≡ k ≡ k ≡
253 (mod 1768) 3 k ≡
695 (mod 1768) 4 k ≡
644 (mod 884) 1 k ≡
865 (mod 884) 2 k ≡
202 (mod 884) 3 k ≡
423 (mod 884) 4 k ≡
151 (mod 663) 1 k ≡
593 (mod 663) 2 k ≡
100 (mod 663) 3 k ≡ k ≡
321 (mod 1326) 2 k ≡ k ≡ d = − congruence pk ≡
542 (mod 2652) 3 k ≡ k ≡
270 (mod 1105) 1 k ≡
712 (mod 1105) 2 k ≡
933 (mod 1105) 3 k ≡
49 (mod 1105) 4 k ≡ k ≡ k ≡ k ≡
661 (mod 3315) 4 k ≡ k ≡
440 (mod 2210) 1 k ≡ k ≡
882 (mod 2210) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
219 (mod 6630) 1 k ≡ k ≡ k ≡
168 (mod 1547) 1 k ≡
610 (mod 1547) 2 k ≡ k ≡ k ≡
389 (mod 3094) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
16 (mod 289) 1 k ≡
33 (mod 289) 2 k ≡
50 (mod 289) 3 k ≡
356 (mod 578) 1 k ≡
67 (mod 578) 2 k ≡
84 (mod 578) 3 k ≡
373 (mod 578) 4 k ≡
390 (mod 578) 5 k ≡ k ≡ k ≡
101 (mod 2312) 3 k ≡
679 (mod 2312) 4 congruence pk ≡ k ≡
696 (mod 1156) 1 k ≡
985 (mod 1156) 2 k ≡
118 (mod 1156) 3 k ≡
407 (mod 1156) 4 k ≡ k ≡
424 (mod 867) 1 k ≡
713 (mod 867) 2 k ≡
730 (mod 867) 3 k ≡
152 (mod 867) 4 k ≡
169 (mod 867) 6 k ≡
458 (mod 867) 5 k ≡
475 (mod 867) 7 k ≡
764 (mod 867) 8 k ≡
135 (mod 1734) 1 k ≡
441 (mod 1734) 2 k ≡
747 (mod 1734) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
781 (mod 2601) 3 k ≡ k ≡ k ≡
203 (mod 5202) 2 k ≡ k ≡ k ≡
220 (mod 1445) 3 k ≡ k ≡ k ≡
798 (mod 1445) 4 k ≡
509 (mod 1445) 5 k ≡
815 (mod 1445) 6 k ≡ k ≡
526 (mod 2890) 2 k ≡ k ≡ k ≡
237 (mod 5780) 10 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ pk ≡
543 (mod 6936) 1 k ≡ k ≡ k ≡
832 (mod 3468) 1 k ≡ k ≡ k ≡
254 (mod 3468) 4 k ≡
560 (mod 2023) 1 k ≡ k ≡
849 (mod 2023) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
288 (mod 3179) 3 k ≡ k ≡ k ≡ k ≡ k ≡
577 (mod 6358) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
866 (mod 9537) 3 k ≡ k ≡ k ≡ k ≡ d = − congruence pk ≡ k ≡
89 (mod 90) 2 k ≡ k ≡
14 (mod 28) 1 k ≡ k ≡
17 (mod 18) 1 k ≡ k ≡ k ≡
98 (mod 99) 1 k ≡
22 (mod 30) 1 k ≡ k ≡
12 (mod 30) 3 k ≡ k ≡
13 (mod 20) 1 k ≡ k ≡ k ≡ k ≡
11 (mod 25) 3 k ≡
66 (mod 75) 1 k ≡
16 (mod 75) 2 k ≡
41 (mod 75) 3 k ≡
21 (mod 125) 1 k ≡
46 (mod 125) 2 k ≡
71 (mod 125) 3 k ≡
96 (mod 125) 4 k ≡
246 (mod 250) 1 k ≡
121 (mod 250) 2 k ≡
25 (mod 50) 1 k ≡ k ≡
35 (mod 50) 3 k ≡
65 (mod 100) 1 k ≡
15 (mod 100) 2 k ≡
45 (mod 100) 3 k ≡
95 (mod 100) 4 k ≡
149 (mod 180) 1 k ≡
59 (mod 180) 2 k ≡
29 (mod 180) 3 k ≡
659 (mod 720) 1 k ≡
299 (mod 720) 2 k ≡
839 (mod 1440) 1 k ≡ k ≡
119 (mod 1440) 3 k ≡
479 (mod 1440) 4 k ≡
19 (mod 360) 1 k ≡
139 (mod 360) 2 k ≡
259 (mod 360) 3 k ≡
679 (mod 840) 1 k ≡
799 (mod 840) 2 k ≡
79 (mod 840) 3 k ≡
199 (mod 840) 4 congruence pk ≡
319 (mod 840) 5 k ≡
439 (mod 1680) 1 k ≡ k ≡ k ≡
559 (mod 1680) 4 k ≡
49 (mod 280) 1 k ≡
189 (mod 280) 2 k ≡
64 (mod 315) 1 k ≡
274 (mod 315) 2 k ≡
169 (mod 315) 3 k ≡
289 (mod 630) 1 k ≡
499 (mod 630) 2 k ≡
79 (mod 630) 3 k ≡
199 (mod 630) 4 k ≡
409 (mod 630) 5 k ≡ k ≡
109 (mod 1260) 2 k ≡
949 (mod 1260) 3 k ≡
529 (mod 1260) 4 k ≡
649 (mod 1260) 5 k ≡
229 (mod 1260) 6 k ≡ k ≡
104 (mod 175) 1 k ≡
34 (mod 175) 2 k ≡
139 (mod 175) 3 k ≡
69 (mod 350) 1 k ≡
349 (mod 350) 2 k ≡ k ≡
28 (mod 40) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
34 (mod 58) 1 k ≡ k ≡
36 (mod 116) 1 k ≡ k ≡
96 (mod 116) 3 k ≡
68 (mod 116) 4 k ≡
210 (mod 232) 1 k ≡
66 (mod 232) 2 k ≡
154 (mod 232) 3 k ≡
10 (mod 232) 4 k ≡
40 (mod 232) 5 k ≡
94 (mod 464) 1 k ≡
414 (mod 464) 2 k ≡
270 (mod 464) 3 k ≡
126 (mod 464) 4 k ≡
98 (mod 464) 5 congruence pk ≡
330 (mod 464) 6 k ≡
446 (mod 464) 7 k ≡
388 (mod 928) 1 k ≡
620 (mod 928) 2 k ≡
852 (mod 928) 3 k ≡
156 (mod 928) 4 k ≡
12 (mod 87) 1 k ≡
70 (mod 87) 2 k ≡
41 (mod 87) 3 k ≡
42 (mod 174) 1 k ≡
100 (mod 174) 2 k ≡
158 (mod 174) 3 k ≡
72 (mod 348) 1 k ≡
304 (mod 348) 2 k ≡
188 (mod 348) 3 k ≡
16 (mod 348) 4 k ≡
190 (mod 348) 5 k ≡
594 (mod 696) 1 k ≡
130 (mod 696) 2 k ≡
362 (mod 696) 3 k ≡
942 (mod 1392) 1 k ≡
478 (mod 1392) 2 k ≡
14 (mod 1392) 3 k ≡
189 (mod 261) 1 k ≡
73 (mod 261) 2 k ≡
218 (mod 261) 3 k ≡
102 (mod 261) 4 k ≡
247 (mod 261) 5 k ≡
131 (mod 261) 6 k ≡
15 (mod 261) 7 k ≡
160 (mod 261) 8 k ≡
44 (mod 522) 1 k ≡
306 (mod 522) 2 k ≡
480 (mod 522) 3 k ≡
132 (mod 522) 4 k ≡
596 (mod 1044) 1 k ≡
74 (mod 1044) 2 k ≡
248 (mod 1044) 4 k ≡
770 (mod 1044) 5 k ≡
944 (mod 1044) 3 k ≡
422 (mod 1044) 6 k ≡
48 (mod 60) 1 k ≡
28 (mod 60) 2 k ≡ k ≡
75 (mod 145) 1 k ≡
104 (mod 145) 2 k ≡
105 (mod 145) 3 k ≡
134 (mod 145) 4 k ≡
280 (mod 290) 1 k ≡
164 (mod 290) 2 d = − congruence pk ≡
20 (mod 290) 3 k ≡
194 (mod 290) 4 k ≡
50 (mod 290) 5 k ≡
224 (mod 290) 6 k ≡
278 (mod 580) 1 k ≡
18 (mod 580) 2 k ≡
338 (mod 580) 3 k ≡
78 (mod 580) 4 k ≡
978 (mod 1160) 1 k ≡
398 (mod 1160) 2 k ≡
225 (mod 435) 1 k ≡
370 (mod 435) 2 k ≡
80 (mod 435) 3 k ≡
399 (mod 435) 4 k ≡
544 (mod 870) 1 k ≡
254 (mod 870) 2 k ≡
690 (mod 870) 3 k ≡
400 (mod 870) 4 k ≡
110 (mod 870) 5 k ≡
864 (mod 870) 6 k ≡
574 (mod 870) 7 k ≡ k ≡ k ≡
284 (mod 3480) 3 k ≡ k ≡ k ≡
138 (mod 1740) 1 k ≡
718 (mod 1740) 2 k ≡ k ≡ k ≡ k ≡
458 (mod 1740) 6 k ≡
198 (mod 1740) 7 k ≡
778 (mod 1740) 8 k ≡ k ≡
140 (mod 1015) 1 congruence pk ≡
575 (mod 1015) 2 k ≡ k ≡
430 (mod 1015) 4 k ≡
865 (mod 1015) 5 k ≡
285 (mod 1015) 6 k ≡
720 (mod 1015) 7 k ≡ k ≡ k ≡
604 (mod 2030) 3 k ≡
24 (mod 2030) 4 k ≡ k ≡ k ≡ k ≡
894 (mod 4060) 4 k ≡ k ≡
314 (mod 4060) 6 k ≡
141 (mod 203) 1 k ≡
170 (mod 203) 2 k ≡
199 (mod 203) 3 k ≡
25 (mod 203) 4 k ≡
54 (mod 406) 1 k ≡
286 (mod 406) 2 k ≡
316 (mod 406) 3 k ≡
142 (mod 406) 4 k ≡
374 (mod 406) 5 k ≡
112 (mod 812) 1 k ≡
84 (mod 812) 2 k ≡
200 (mod 812) 3 k ≡
606 (mod 812) 4 k ≡
432 (mod 812) 5 k ≡
26 (mod 812) 6 k ≡
258 (mod 609) 1 k ≡
55 (mod 609) 2 k ≡
461 (mod 609) 3 k ≡
143 (mod 319) 1 k ≡
56 (mod 319) 2 congruence pk ≡
288 (mod 319) 3 k ≡
201 (mod 319) 4 k ≡
346 (mod 638) 1 k ≡
578 (mod 638) 2 k ≡
172 (mod 638) 3 k ≡
404 (mod 638) 4 k ≡
636 (mod 638) 5 k ≡
114 (mod 1276) 1 k ≡
868 (mod 1276) 2 k ≡
230 (mod 1276) 3 k ≡
260 (mod 377) 1 k ≡
144 (mod 377) 2 k ≡
28 (mod 377) 3 k ≡
289 (mod 377) 4 k ≡
550 (mod 754) 1 k ≡
434 (mod 754) 2 k ≡ k ≡
318 (mod 1508) 1 k ≡
956 (mod 1508) 2 k ≡
202 (mod 1508) 4 k ≡
840 (mod 1131) 1 k ≡
463 (mod 1131) 2 k ≡
86 (mod 1131) 3 k ≡ k ≡
724 (mod 2262) 1 k ≡ k ≡
985 (mod 1885) 1 k ≡
608 (mod 1885) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ Table 8: Covering information for d = − congruence pk ≡ k ≡ k ≡ k ≡
16 (mod 26) 1 k ≡ k ≡ k ≡
17 (mod 52) 2 k ≡
82 (mod 208) 1 k ≡
134 (mod 208) 2 congruence pk ≡
186 (mod 208) 3 k ≡
30 (mod 208) 4 k ≡
147 (mod 208) 5 k ≡
199 (mod 208) 6 k ≡
43 (mod 416) 1 k ≡
251 (mod 416) 2 k ≡
303 (mod 416) 3 k ≡
95 (mod 416) 4 k ≡
18 (mod 39) 1 congruence pk ≡
70 (mod 78) 1 k ≡
31 (mod 78) 2 k ≡
44 (mod 78) 3 k ≡ k ≡ k ≡
45 (mod 117) 1 k ≡
84 (mod 117) 2 k ≡ k ≡
58 (mod 234) 1 d = − congruence pk ≡
136 (mod 234) 2 k ≡
214 (mod 234) 3 k ≡
110 (mod 117) 4 k ≡
32 (mod 351) 1 k ≡
149 (mod 351) 2 k ≡
266 (mod 351) 3 k ≡
305 (mod 351) 4 k ≡
71 (mod 702) 1 k ≡
188 (mod 702) 2 k ≡
422 (mod 702) 3 k ≡
539 (mod 702) 4 k ≡
20 (mod 65) 1 k ≡
46 (mod 65) 2 k ≡ k ≡
72 (mod 130) 2 k ≡
33 (mod 390) 1 k ≡
163 (mod 390) 2 k ≡
293 (mod 390) 3 k ≡
228 (mod 520) 1 k ≡
488 (mod 520) 2 k ≡
98 (mod 1040) 1 k ≡
358 (mod 1040) 2 k ≡
618 (mod 1040) 3 k ≡
878 (mod 1040) 4 k ≡
59 (mod 195) 1 k ≡
124 (mod 195) 2 k ≡
189 (mod 195) 3 k ≡ k ≡
60 (mod 104) 2 k ≡
21 (mod 156) 1 k ≡
73 (mod 156) 2 k ≡
125 (mod 156) 3 k ≡
34 (mod 312) 1 k ≡
86 (mod 312) 2 k ≡
138 (mod 312) 3 k ≡
242 (mod 312) 4 k ≡
294 (mod 312) 5 k ≡
190 (mod 624) 1 k ≡
502 (mod 624) 2 k ≡
47 (mod 260) 1 k ≡
99 (mod 260) 2 k ≡
151 (mod 260) 3 k ≡
203 (mod 260) 4 k ≡
255 (mod 260) 5 k ≡ k ≡
22 (mod 91) 2 k ≡
35 (mod 91) 3 k ≡
48 (mod 91) 4 k ≡
61 (mod 91) 5 k ≡
74 (mod 91) 6 congruence pk ≡
87 (mod 91) 7 k ≡
49 (mod 182) 1 k ≡
140 (mod 182) 2 k ≡
36 (mod 364) 1 k ≡
127 (mod 364) 2 k ≡
309 (mod 364) 3 k ≡
218 (mod 728) 1 k ≡
582 (mod 728) 2 k ≡
23 (mod 273) 1 k ≡
114 (mod 273) 2 k ≡
205 (mod 273) 3 k ≡
10 (mod 273) 4 k ≡
192 (mod 273) 5 k ≡
101 (mod 546) 1 k ≡
374 (mod 546) 2 k ≡
88 (mod 455) 1 k ≡
179 (mod 455) 2 k ≡
270 (mod 455) 3 k ≡
361 (mod 455) 4 k ≡
452 (mod 455) 5 k ≡
75 (mod 910) 1 k ≡
257 (mod 910) 2 k ≡
439 (mod 910) 3 k ≡
621 (mod 910) 4 k ≡
803 (mod 910) 5 k ≡
348 (mod 1456) 1 k ≡
530 (mod 1456) 2 k ≡
712 (mod 1456) 3 k ≡ k ≡ k ≡ k ≡
166 (mod 2912) 1 k ≡ k ≡ k ≡
62 (mod 416) 5 k ≡
244 (mod 1001) 1 k ≡
517 (mod 1001) 2 k ≡
608 (mod 1001) 3 k ≡
881 (mod 1001) 4 k ≡
335 (mod 2002) 1 k ≡
972 (mod 2002) 2 k ≡ k ≡ k ≡ k ≡
699 (mod 4004) 1 k ≡ k ≡
62 (mod 3003) 1 k ≡ k ≡ k ≡
426 (mod 6006) 1 congruence pk ≡ k ≡ k ≡ k ≡ k ≡ k ≡
790 (mod 5005) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
153 (mod 10010) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
11 (mod 143) 1 k ≡
24 (mod 143) 2 k ≡
89 (mod 143) 3 k ≡
102 (mod 143) 4 k ≡
37 (mod 286) 1 k ≡
115 (mod 286) 2 k ≡
180 (mod 286) 3 k ≡
258 (mod 286) 4 k ≡
50 (mod 429) 1 k ≡
193 (mod 429) 2 k ≡
336 (mod 429) 3 k ≡
128 (mod 572) 1 k ≡
271 (mod 572) 2 k ≡
414 (mod 572) 3 k ≡
557 (mod 572) 4 k ≡
63 (mod 1716) 1 k ≡
635 (mod 1716) 2 k ≡ k ≡
349 (mod 572) 5 d = − congruence pk ≡
206 (mod 286) 5 k ≡
141 (mod 858) 1 k ≡
284 (mod 858) 2 k ≡
427 (mod 858) 3 k ≡
570 (mod 858) 4 k ≡
856 (mod 858) 5 k ≡
713 (mod 1716) 4 k ≡ k ≡ k ≡
76 (mod 715) 1 k ≡
362 (mod 715) 2 k ≡
505 (mod 715) 3 k ≡
219 (mod 1430) 1 k ≡
648 (mod 1430) 2 k ≡
934 (mod 1430) 3 k ≡ k ≡
12 (mod 169) 1 k ≡
25 (mod 169) 2 k ≡
38 (mod 169) 3 k ≡
51 (mod 338) 1 k ≡
64 (mod 338) 2 k ≡
220 (mod 338) 3 k ≡
233 (mod 338) 4 k ≡
77 (mod 507) 2 k ≡
90 (mod 507) 1 k ≡
246 (mod 507) 3 k ≡
259 (mod 507) 4 k ≡
415 (mod 507) 5 k ≡
428 (mod 507) 6 k ≡
441 (mod 507) 7 k ≡
103 (mod 1014) 1 congruence pk ≡
272 (mod 1014) 2 k ≡
610 (mod 1014) 3 k ≡
779 (mod 1014) 4 k ≡
116 (mod 676) 1 k ≡
285 (mod 676) 2 k ≡
454 (mod 676) 3 k ≡
623 (mod 2704) 1 k ≡ k ≡ k ≡ k ≡
129 (mod 1352) 1 k ≡
298 (mod 1352) 2 k ≡
467 (mod 1352) 3 k ≡
636 (mod 1352) 4 k ≡
805 (mod 1352) 5 k ≡
974 (mod 1352) 6 k ≡ k ≡ k ≡
311 (mod 845) 1 k ≡
480 (mod 845) 2 k ≡
142 (mod 1690) 1 k ≡
818 (mod 1690) 2 k ≡
987 (mod 1690) 3 k ≡ k ≡
649 (mod 3380) 1 k ≡ k ≡ k ≡ k ≡
155 (mod 1183) 1 k ≡
324 (mod 1183) 2 k ≡
493 (mod 1183) 3 congruence pk ≡ k ≡
662 (mod 2366) 1 k ≡
831 (mod 2366) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
168 (mod 2535) 1 k ≡
675 (mod 2535) 2 k ≡ k ≡ k ≡ k ≡ k ≡
506 (mod 1521) 1 k ≡ k ≡ k ≡ k ≡ k ≡
844 (mod 10140) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ Table 9: Covering information for d = − congruence pk ≡ k ≡ k ≡ k ≡
11 (mod 21) 1 k ≡
14 (mod 22) 1 k ≡
40 (mod 44) 1 k ≡
96 (mod 99) 1 k ≡ k ≡
24 (mod 64) 2 k ≡
40 (mod 64) 3 k ≡
56 (mod 64) 4 k ≡
25 (mod 210) 1 k ≡
151 (mod 210) 2 k ≡
67 (mod 210) 3 congruence pk ≡
193 (mod 420) 1 k ≡
403 (mod 420) 2 k ≡
109 (mod 420) 3 k ≡
319 (mod 420) 4 k ≡
18 (mod 336) 1 k ≡
228 (mod 336) 2 k ≡
102 (mod 336) 3 k ≡
522 (mod 672) 1 k ≡
186 (mod 672) 2 k ≡
396 (mod 672) 3 k ≡
60 (mod 672) 4 k ≡
270 (mod 672) 5 k ≡
606 (mod 672) 6 k ≡
81 (mod 126) 1 congruence pk ≡
39 (mod 126) 2 k ≡
249 (mod 252) 1 k ≡
123 (mod 252) 2 k ≡
113 (mod 176) 1 k ≡
69 (mod 176) 2 k ≡
201 (mod 352) 1 k ≡
333 (mod 352) 2 k ≡
25 (mod 352) 3 k ≡
157 (mod 352) 4 k ≡ k ≡
531 (mod 704) 2 k ≡
355 (mod 704) 3 k ≡
179 (mod 704) 4 k ≡
267 (mod 1056) 1 d = − congruence pk ≡
619 (mod 1056) 2 k ≡
971 (mod 1056) 3 k ≡
795 (mod 2112) 1 k ≡
91 (mod 2112) 2 k ≡ k ≡ k ≡ k ≡
443 (mod 4224) 1 k ≡ k ≡
135 (mod 440) 1 k ≡
311 (mod 440) 2 k ≡
487 (mod 880) 1 k ≡
47 (mod 880) 2 k ≡
663 (mod 880) 3 k ≡
223 (mod 880) 4 k ≡
839 (mod 880) 5 k ≡
399 (mod 1760) 1 k ≡ k ≡ k ≡
99 (mod 231) 2 k ≡
198 (mod 231) 3 k ≡
66 (mod 231) 4 k ≡
33 (mod 231) 5 k ≡
132 (mod 231) 6 k ≡
385 (mod 462) 1 k ≡
253 (mod 462) 2 k ≡
121 (mod 462) 3 k ≡
770 (mod 924) 1 k ≡
638 (mod 924) 2 k ≡
506 (mod 924) 3 k ≡
374 (mod 924) 4 k ≡
110 (mod 924) 5 k ≡
308 (mod 1848) 1 k ≡ k ≡
44 (mod 1848) 3 k ≡
836 (mod 1848) 4 k ≡
451 (mod 693) 1 k ≡
220 (mod 693) 2 k ≡
682 (mod 693) 3 k ≡
649 (mod 693) 4 k ≡
418 (mod 693) 5 k ≡
187 (mod 693) 6 k ≡
55 (mod 1386) 1 k ≡
979 (mod 1386) 2 k ≡
517 (mod 1386) 3 k ≡ k ≡ k ≡ k ≡
902 (mod 3696) 4 k ≡ pk ≡ k ≡ k ≡
572 (mod 7392) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
58 (mod 76) 1 k ≡
58 (mod 76) 2 k ≡
20 (mod 152) 1 k ≡
116 (mod 152) 2 k ≡ k ≡
78 (mod 152) 4 k ≡
97 (mod 304) 1 k ≡
211 (mod 304) 2 k ≡
21 (mod 304) 3 k ≡
135 (mod 608) 1 k ≡
439 (mod 608) 2 k ≡
553 (mod 608) 3 k ≡
857 (mod 1216) 1 k ≡
249 (mod 1216) 2 k ≡
971 (mod 1216) 3 k ≡
667 (mod 1216) 4 k ≡
363 (mod 1216) 5 k ≡
59 (mod 1216) 7 k ≡
781 (mod 1216) 6 k ≡
477 (mod 1216) 8 k ≡
173 (mod 1216) 9 k ≡ k ≡ k ≡ k ≡
63 (mod 192) 4 k ≡
159 (mod 384) 1 k ≡
31 (mod 384) 2 k ≡
303 (mod 384) 3 k ≡
127 (mod 384) 4 k ≡
319 (mod 384) 5 k ≡
559 (mod 768) 1 k ≡
175 (mod 768) 2 k ≡
591 (mod 768) 3 k ≡
79 (mod 768) 4 k ≡
207 (mod 768) 5 k ≡
463 (mod 768) 6 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
42 (mod 57) 2 congruence pk ≡
24 (mod 57) 3 k ≡
79 (mod 114) 1 k ≡
98 (mod 114) 2 k ≡
61 (mod 228) 1 k ≡
175 (mod 228) 2 k ≡
194 (mod 228) 3 k ≡
157 (mod 228) 4 k ≡
43 (mod 228) 5 k ≡
62 (mod 228) 6 k ≡
308 (mod 456) 1 k ≡
404 (mod 456) 2 k ≡
25 (mod 456) 3 k ≡
367 (mod 456) 4 k ≡
253 (mod 456) 5 k ≡
63 (mod 171) 1 k ≡
120 (mod 171) 2 k ≡ k ≡
595 (mod 912) 1 k ≡
139 (mod 912) 2 k ≡
956 (mod 1368) 1 k ≡
44 (mod 1368) 2 k ≡
500 (mod 1368) 3 k ≡ k ≡ k ≡ k ≡
45 (mod 342) 1 k ≡
216 (mod 342) 2 k ≡
273 (mod 342) 3 k ≡
102 (mod 342) 4 k ≡
159 (mod 342) 5 k ≡
330 (mod 3420) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
577 (mod 684) 1 k ≡ k ≡
121 (mod 684) 3 k ≡
235 (mod 684) 4 k ≡ k ≡
349 (mod 2736) 1 k ≡ k ≡
463 (mod 2736) 3 k ≡ k ≡ d = − congruence pk ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
140 (mod 570) 1 k ≡
26 (mod 570) 2 k ≡
482 (mod 570) 3 k ≡
368 (mod 570) 4 k ≡
254 (mod 1140) 1 k ≡
824 (mod 1140) 2 k ≡
65 (mod 95) 1 k ≡
46 (mod 95) 2 k ≡
27 (mod 95) 3 k ≡ k ≡
84 (mod 95) 5 k ≡
66 (mod 285) 1 k ≡
256 (mod 285) 2 k ≡
161 (mod 285) 3 k ≡
142 (mod 190) 1 k ≡
123 (mod 190) 2 k ≡
104 (mod 190) 3 k ≡
85 (mod 380) 1 k ≡
275 (mod 380) 2 k ≡
47 (mod 380) 3 k ≡
237 (mod 380) 4 k ≡
218 (mod 380) 5 k ≡ k ≡
199 (mod 380) 7 k ≡
370 (mod 380) 8 k ≡
180 (mod 760) 1 k ≡
28 (mod 760) 2 k ≡
145 (mod 1140) 3 k ≡
715 (mod 1140) 5 k ≡
601 (mod 1140) 6 k ≡
31 (mod 1140) 4 k ≡ k ≡
487 (mod 1140) 7 k ≡
373 (mod 1140) 9 k ≡
943 (mod 1140) 10 k ≡
829 (mod 1140) 11 k ≡
259 (mod 2280) 1 k ≡ k ≡
105 (mod 133) 1 k ≡
29 (mod 133) 2 k ≡
86 (mod 133) 3 k ≡
143 (mod 266) 1 k ≡
257 (mod 266) 2 k ≡
181 (mod 266) 3 congruence pk ≡
10 (mod 532) 1 k ≡
390 (mod 532) 2 k ≡
314 (mod 532) 3 k ≡
49 (mod 532) 4 k ≡
182 (mod 532) 5 k ≡
276 (mod 1064) 1 k ≡
124 (mod 1064) 2 k ≡
580 (mod 1064) 3 k ≡ k ≡ k ≡
315 (mod 2128) 3 k ≡
847 (mod 2128) 4 k ≡ k ≡ k ≡
980 (mod 4256) 3 k ≡ k ≡
505 (mod 665) 1 k ≡
106 (mod 665) 2 k ≡
372 (mod 665) 3 k ≡ k ≡
239 (mod 1330) 2 k ≡ k ≡
638 (mod 2660) 1 k ≡ k ≡ k ≡ k ≡
30 (mod 1995) 1 k ≡ k ≡ k ≡
828 (mod 1995) 4 k ≡ k ≡
961 (mod 3990) 2 k ≡ k ≡ k ≡ k ≡
163 (mod 7980) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
429 (mod 5985) 3 k ≡
486 (mod 1197) 1 congruence pk ≡
885 (mod 1197) 2 k ≡
87 (mod 1197) 3 k ≡ k ≡ k ≡
619 (mod 2394) 3 k ≡ k ≡
752 (mod 2394) 5 k ≡
201 (mod 931) 1 k ≡
600 (mod 931) 2 k ≡
68 (mod 931) 3 k ≡
467 (mod 931) 4 k ≡
866 (mod 931) 5 k ≡
334 (mod 931) 6 k ≡
733 (mod 931) 7 k ≡
790 (mod 1862) 1 k ≡ k ≡ k ≡ k ≡ k ≡
258 (mod 3724) 5 k ≡ k ≡
657 (mod 7448) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
391 (mod 5586) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
524 (mod 22344) 1 k ≡ k ≡ k ≡
126 (mod 399) 1 k ≡
183 (mod 399) 2 k ≡
240 (mod 399) 3 k ≡
297 (mod 399) 4 k ≡
12 (mod 399) 5 k ≡
468 (mod 798) 1 d = − congruence pk ≡
69 (mod 798) 2 k ≡
734 (mod 798) 3 k ≡ k ≡
50 (mod 1596) 2 k ≡
506 (mod 1596) 3 k ≡
962 (mod 1596) 4 k ≡ k ≡ k ≡ k ≡
164 (mod 6384) 1 k ≡ k ≡ k ≡ k ≡
278 (mod 4788) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
108 (mod 513) 1 k ≡
279 (mod 513) 2 k ≡
963 (mod 1026) 1 k ≡
165 (mod 1026) 2 k ≡
849 (mod 1026) 3 k ≡
507 (mod 1026) 4 k ≡
18 (mod 108) 1 k ≡
30 (mod 108) 2 k ≡
66 (mod 108) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
60 (mod 135) 1 k ≡ k ≡
87 (mod 135) 3 k ≡
33 (mod 135) 4 k ≡
114 (mod 135) 5 k ≡ k ≡
906 (mod 2565) 2 k ≡ k ≡
393 (mod 2565) 4 k ≡
69 (mod 270) 1 k ≡
105 (mod 270) 2 k ≡
51 (mod 270) 3 k ≡
267 (mod 270) 4 k ≡ pk ≡ k ≡
474 (mod 540) 1 k ≡
510 (mod 540) 2 k ≡
186 (mod 540) 3 k ≡
402 (mod 540) 4 k ≡
78 (mod 540) 5 k ≡
294 (mod 540) 6 k ≡
204 (mod 540) 7 k ≡ k ≡ k ≡ k ≡ k ≡
564 (mod 10260) 5 k ≡ k ≡
38 (mod 96) 1 k ≡
86 (mod 96) 2 k ≡
74 (mod 96) 3 k ≡
26 (mod 96) 4 k ≡
14 (mod 192) 1 k ≡
158 (mod 192) 2 k ≡
110 (mod 192) 3 k ≡
350 (mod 480) 1 k ≡
446 (mod 480) 2 k ≡
62 (mod 480) 3 k ≡
638 (mod 960) 1 k ≡
254 (mod 960) 2 k ≡ k ≡
241 (mod 1824) 2 k ≡ k ≡
469 (mod 1824) 4 k ≡ k ≡ k ≡
697 (mod 3648) 2 k ≡
45 (mod 80) 1 k ≡
61 (mod 80) 2 k ≡
157 (mod 240) 1 k ≡
13 (mod 240) 2 k ≡
109 (mod 240) 3 k ≡
127 (mod 228) 7 k ≡
20 (mod 120) 1 k ≡
716 (mod 2280) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
166 (mod 209) 1 k ≡
90 (mod 209) 2 k ≡
147 (mod 209) 3 k ≡
71 (mod 209) 4 congruence pk ≡
204 (mod 209) 5 k ≡
337 (mod 418) 1 k ≡
261 (mod 418) 2 k ≡
185 (mod 418) 3 k ≡
109 (mod 418) 4 k ≡
546 (mod 836) 1 k ≡
470 (mod 836) 2 k ≡
394 (mod 836) 3 k ≡ k ≡ k ≡ k ≡
318 (mod 3344) 4 k ≡
964 (mod 1672) 1 k ≡
52 (mod 1672) 2 k ≡
812 (mod 1672) 3 k ≡ k ≡
243 (mod 627) 1 k ≡
585 (mod 627) 2 k ≡
15 (mod 627) 3 k ≡
357 (mod 627) 4 k ≡
72 (mod 2508) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
699 (mod 20064) 4 k ≡ k ≡
452 (mod 5016) 2 k ≡ k ≡ k ≡ k ≡
908 (mod 5016) 6 k ≡
661 (mod 1254) 1 k ≡ k ≡
433 (mod 1254) 3 k ≡
775 (mod 1254) 4 k ≡ k ≡
205 (mod 1045) 1 k ≡ k ≡
832 (mod 1045) 3 k ≡
623 (mod 2090) 1 k ≡ k ≡
965 (mod 2090) 3 k ≡ k ≡ k ≡ d = − congruence pk ≡
414 (mod 4180) 4 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
756 (mod 16720) 1 k ≡ k ≡ k ≡ k ≡
129 (mod 3135) 5 k ≡ k ≡
471 (mod 3135) 3 k ≡ k ≡ k ≡ k ≡ k ≡
547 (mod 6270) 1 k ≡ k ≡ k ≡
625 (mod 660) 1 k ≡
295 (mod 660) 2 k ≡
361 (mod 660) 5 k ≡
31 (mod 660) 3 k ≡
97 (mod 660) 4 k ≡
427 (mod 660) 6 k ≡
493 (mod 660) 7 k ≡
163 (mod 660) 8 k ≡
889 (mod 12540) 1 k ≡ k ≡ k ≡ k ≡
20 (mod 1320) 1 k ≡ k ≡
812 (mod 1320) 3 k ≡
548 (mod 1320) 4 k ≡
284 (mod 1320) 5 k ≡
932 (mod 1320) 6 k ≡
668 (mod 1320) 7 k ≡
404 (mod 1320) 8 k ≡ k ≡
351 (mod 495) 1 k ≡
21 (mod 495) 2 k ≡
186 (mod 495) 3 k ≡
252 (mod 495) 4 k ≡
417 (mod 495) 5 k ≡
87 (mod 495) 6 k ≡ k ≡
813 (mod 9405) 4 k ≡ pk ≡ k ≡ k ≡ k ≡
901 (mod 990) 1 k ≡
571 (mod 990) 2 k ≡
241 (mod 990) 3 k ≡
307 (mod 990) 4 k ≡ k ≡ k ≡ k ≡
373 (mod 1980) 1 k ≡ k ≡ k ≡
43 (mod 1980) 5 k ≡
109 (mod 1980) 4 k ≡ k ≡
769 (mod 1980) 7 k ≡ k ≡ k ≡ k ≡
956 (mod 3960) 1 k ≡ k ≡ k ≡ k ≡
692 (mod 3960) 5 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
130 (mod 247) 1 k ≡
92 (mod 247) 3 k ≡
54 (mod 247) 2 k ≡
16 (mod 247) 4 k ≡
225 (mod 247) 5 k ≡
187 (mod 494) 1 k ≡
149 (mod 494) 2 k ≡
111 (mod 494) 3 k ≡
73 (mod 494) 4 k ≡
434 (mod 988) 1 k ≡
890 (mod 988) 2 k ≡
358 (mod 988) 3 k ≡ k ≡
814 (mod 2964) 2 k ≡ k ≡ k ≡
396 (mod 1976) 2 k ≡
852 (mod 1976) 3 congruence pk ≡ k ≡
282 (mod 741) 1 k ≡
529 (mod 741) 2 k ≡
35 (mod 741) 3 k ≡
738 (mod 741) 4 k ≡
453 (mod 741) 5 k ≡
985 (mod 1482) 1 k ≡ k ≡ k ≡ k ≡
415 (mod 1235) 1 k ≡ k ≡
662 (mod 1235) 3 k ≡
168 (mod 1235) 4 k ≡
909 (mod 1235) 5 k ≡
29 (mod 34) 1 k ≡
128 (mod 204) 1 k ≡
17 (mod 323) 1 k ≡
188 (mod 323) 2 k ≡
36 (mod 323) 3 k ≡
530 (mod 646) 1 k ≡
207 (mod 646) 2 k ≡
55 (mod 646) 3 k ≡
549 (mod 646) 4 k ≡
397 (mod 646) 5 k ≡
245 (mod 646) 6 k ≡
93 (mod 646) 7 k ≡
378 (mod 1292) 1 k ≡
226 (mod 1292) 2 k ≡
74 (mod 1292) 3 k ≡
330 (mod 340) 1 k ≡
126 (mod 340) 2 k ≡
262 (mod 340) 3 k ≡
58 (mod 340) 4 k ≡
194 (mod 340) 5 k ≡ k ≡ k ≡ k ≡
454 (mod 2584) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
264 (mod 969) 1 d = − congruence pk ≡
435 (mod 969) 2 k ≡
606 (mod 969) 3 k ≡
948 (mod 969) 4 k ≡ k ≡ k ≡
150 (mod 2907) 3 k ≡ k ≡
321 (mod 3876) 2 k ≡ k ≡ k ≡ k ≡ k ≡
283 (mod 1938) 3 k ≡
625 (mod 1938) 4 k ≡ k ≡ k ≡
967 (mod 5814) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
644 (mod 7752) 5 k ≡
815 (mod 1615) 1 k ≡ k ≡
492 (mod 3230) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
169 (mod 6460) 4 k ≡ pk ≡ k ≡
18 (mod 361) 1 k ≡
37 (mod 361) 3 k ≡
56 (mod 361) 2 k ≡
75 (mod 361) 4 k ≡
94 (mod 361) 5 k ≡
113 (mod 361) 6 k ≡
132 (mod 361) 7 k ≡
151 (mod 722) 1 k ≡
531 (mod 722) 2 k ≡
189 (mod 722) 3 k ≡ k ≡
170 (mod 1444) 2 k ≡
550 (mod 1444) 3 k ≡
512 (mod 1444) 4 k ≡
892 (mod 2888) 1 k ≡ k ≡ k ≡
930 (mod 1083) 1 k ≡
588 (mod 1083) 2 k ≡
246 (mod 1083) 4 k ≡
987 (mod 1083) 3 k ≡
645 (mod 1083) 5 k ≡
303 (mod 1083) 6 k ≡ k ≡
702 (mod 3249) 1 k ≡ k ≡ k ≡ k ≡
949 (mod 2166) 2 k ≡
607 (mod 2166) 3 k ≡
265 (mod 2166) 4 k ≡ pk ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
284 (mod 12996) 2 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
360 (mod 1805) 1 k ≡
721 (mod 1805) 2 k ≡ k ≡ k ≡ Table 10: Covering information for d = − congruence pk ≡ k ≡ k ≡ k ≡ k ≡ k ≡
70 (mod 345) 2 k ≡
116 (mod 345) 3 k ≡
185 (mod 345) 4 k ≡
231 (mod 345) 5 k ≡
300 (mod 345) 6 k ≡
47 (mod 690) 1 k ≡
93 (mod 690) 2 k ≡
277 (mod 690) 3 congruence pk ≡
507 (mod 690) 4 k ≡
553 (mod 690) 5 k ≡
139 (mod 1380) 1 k ≡
369 (mod 1380) 2 k ≡
599 (mod 1380) 3 k ≡
829 (mod 1380) 4 k ≡ k ≡ k ≡ k ≡ k ≡
27 (mod 46) 3 k ≡
29 (mod 46) 4 k ≡ pk ≡
53 (mod 92) 2 k ≡
77 (mod 92) 3 k ≡
31 (mod 184) 1 k ≡
123 (mod 184) 2 k ≡ k ≡
32 (mod 69) 2 k ≡
55 (mod 69) 3 k ≡
33 (mod 138) 1 k ≡
79 (mod 138) 2 k ≡
125 (mod 138) 3 k ≡
57 (mod 276) 1 k ≡
103 (mod 276) 2 k ≡
149 (mod 276) 3 d = − congruence pk ≡
195 (mod 276) 4 k ≡
241 (mod 276) 5 k ≡
11 (mod 552) 1 k ≡
287 (mod 552) 2 k ≡
12 (mod 115) 1 k ≡
35 (mod 115) 2 k ≡
58 (mod 115) 3 k ≡
81 (mod 115) 4 k ≡
104 (mod 115) 5 k ≡
105 (mod 230) 1 k ≡
151 (mod 230) 2 k ≡
13 (mod 460) 1 k ≡
59 (mod 460) 2 k ≡
197 (mod 460) 3 k ≡
243 (mod 460) 4 k ≡
289 (mod 460) 5 k ≡
427 (mod 460) 6 k ≡
14 (mod 161) 1 k ≡
37 (mod 161) 2 k ≡
60 (mod 161) 3 k ≡
106 (mod 161) 4 k ≡
129 (mod 161) 5 k ≡
313 (mod 1288) 1 k ≡
405 (mod 1288) 2 k ≡
635 (mod 1288) 3 k ≡
957 (mod 1288) 5 k ≡ k ≡ k ≡
83 (mod 644) 1 k ≡
15 (mod 322) 1 k ≡
61 (mod 322) 2 k ≡
107 (mod 322) 3 k ≡
199 (mod 322) 4 k ≡
245 (mod 322) 5 k ≡
291 (mod 322) 6 k ≡
153 (mod 644) 2 k ≡
475 (mod 644) 3 k ≡
85 (mod 483) 1 k ≡
154 (mod 483) 2 k ≡
246 (mod 483) 3 k ≡
315 (mod 483) 4 k ≡
407 (mod 483) 5 k ≡
476 (mod 483) 6 k ≡
177 (mod 483) 7 k ≡
499 (mod 1932) 1 k ≡
821 (mod 1932) 2 k ≡ k ≡ k ≡ k ≡
39 (mod 368) 1 congruence pk ≡
85 (mod 368) 2 k ≡
269 (mod 2576) 1 k ≡
591 (mod 2576) 2 k ≡ k ≡
39 (mod 966) 1 k ≡
361 (mod 966) 2 k ≡
683 (mod 966) 3 k ≡
131 (mod 805) 1 k ≡
292 (mod 805) 2 k ≡
453 (mod 805) 3 k ≡
775 (mod 805) 4 k ≡ k ≡
223 (mod 1610) 2 k ≡
545 (mod 1610) 3 k ≡
867 (mod 1610) 4 k ≡ k ≡ k ≡
63 (mod 207) 1 k ≡
109 (mod 207) 2 k ≡
155 (mod 207) 3 k ≡
201 (mod 414) 1 k ≡
247 (mod 414) 2 k ≡
293 (mod 414) 3 k ≡
339 (mod 414) 4 k ≡
385 (mod 414) 5 k ≡
17 (mod 828) 1 k ≡
431 (mod 828) 2 k ≡
110 (mod 253) 1 k ≡
133 (mod 253) 2 k ≡
156 (mod 253) 3 k ≡
179 (mod 253) 4 k ≡
202 (mod 253) 5 k ≡
225 (mod 506) 1 k ≡
271 (mod 506) 2 k ≡
501 (mod 506) 3 k ≡
41 (mod 1012) 1 k ≡
317 (mod 1012) 2 k ≡
547 (mod 1012) 3 k ≡
823 (mod 2024) 1 k ≡ k ≡
87 (mod 759) 1 k ≡
340 (mod 759) 2 k ≡
593 (mod 759) 3 k ≡
65 (mod 299) 1 k ≡
157 (mod 299) 2 k ≡
249 (mod 299) 3 k ≡
341 (mod 598) 1 k ≡
433 (mod 598) 2 k ≡
525 (mod 598) 3 k ≡
19 (mod 1196) 2 congruence pk ≡
111 (mod 1196) 1 k ≡
203 (mod 1196) 3 k ≡
617 (mod 1196) 4 k ≡
709 (mod 1196) 5 k ≡
801 (mod 1196) 6 k ≡
893 (mod 1196) 7 k ≡
295 (mod 2392) 1 k ≡ k ≡
88 (mod 897) 2 k ≡
387 (mod 897) 3 k ≡
686 (mod 897) 1 k ≡
479 (mod 1794) 1 k ≡
571 (mod 1794) 2 k ≡ k ≡ k ≡ k ≡ k ≡
117 (mod 552) 3 k ≡
393 (mod 552) 4 k ≡
531 (mod 2208) 1 k ≡
807 (mod 2208) 2 k ≡ k ≡ k ≡ k ≡ k ≡
255 (mod 4416) 1 k ≡ k ≡ k ≡ k ≡
25 (mod 1104) 1 k ≡
163 (mod 1104) 2 k ≡
301 (mod 1104) 3 k ≡
439 (mod 1104) 4 k ≡
577 (mod 1104) 5 k ≡
715 (mod 1104) 6 k ≡
853 (mod 1104) 7 k ≡ k ≡ k ≡
991 (mod 2208) 5 k ≡ k ≡
140 (mod 621) 2 k ≡
278 (mod 621) 3 k ≡
416 (mod 621) 4 k ≡
71 (mod 1242) 1 k ≡
209 (mod 1242) 2 k ≡
347 (mod 1242) 3 k ≡ k ≡
485 (mod 2484) 1 k ≡ k ≡
20 (mod 391) 1 d = − congruence pk ≡
204 (mod 391) 2 k ≡
273 (mod 391) 3 k ≡
342 (mod 391) 4 k ≡
43 (mod 782) 1 k ≡
89 (mod 782) 2 k ≡
227 (mod 782) 3 k ≡
365 (mod 782) 4 k ≡
549 (mod 782) 5 k ≡
687 (mod 782) 6 k ≡
181 (mod 1564) 1 k ≡
503 (mod 1564) 2 k ≡
963 (mod 1564) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
319 (mod 3128) 1 k ≡ k ≡ k ≡ k ≡ k ≡
388 (mod 1173) 1 k ≡
779 (mod 1173) 2 k ≡ k ≡
457 (mod 2346) 1 k ≡ k ≡
848 (mod 1173) 4 k ≡
135 (mod 2346) 3 k ≡ k ≡
917 (mod 4692) 1 k ≡ k ≡
21 (mod 437) 1 k ≡
136 (mod 437) 2 congruence pk ≡
228 (mod 437) 3 k ≡
251 (mod 437) 4 k ≡
343 (mod 437) 5 k ≡
159 (mod 874) 1 k ≡
389 (mod 874) 2 k ≡
481 (mod 874) 3 k ≡
711 (mod 874) 4 k ≡
803 (mod 874) 5 k ≡
67 (mod 1748) 1 k ≡
297 (mod 1748) 2 k ≡
619 (mod 1748) 3 k ≡
941 (mod 1748) 4 k ≡ k ≡
849 (mod 3496) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
90 (mod 1311) 1 k ≡
527 (mod 1311) 2 k ≡
964 (mod 1311) 3 k ≡
205 (mod 2622) 1 k ≡
757 (mod 2622) 2 k ≡ k ≡ k ≡ k ≡ k ≡
435 (mod 2185) 1 k ≡
872 (mod 2185) 2 k ≡ k ≡ k ≡ k ≡
113 (mod 4370) 1 congruence pk ≡
987 (mod 4370) 2 k ≡ k ≡ k ≡ k ≡
45 (mod 460) 7 k ≡
275 (mod 460) 8 k ≡
321 (mod 1380) 7 k ≡
781 (mod 1380) 9 k ≡ k ≡
91 (mod 920) 1 k ≡
551 (mod 920) 2 k ≡
137 (mod 1035) 1 k ≡
252 (mod 1035) 2 k ≡
712 (mod 1035) 3 k ≡
367 (mod 2070) 1 k ≡
597 (mod 2070) 2 k ≡
827 (mod 2070) 3 k ≡ k ≡ k ≡ k ≡
68 (mod 575) 1 k ≡
183 (mod 575) 2 k ≡
229 (mod 575) 3 k ≡
298 (mod 575) 4 k ≡
413 (mod 575) 5 k ≡
528 (mod 575) 6 k ≡
459 (mod 1150) 1 k ≡
689 (mod 1150) 2 k ≡
919 (mod 2300) 1 k ≡ k ≡ k ≡ Table 11: Covering information for d = 1 (Part I) congruence pk ≡ k ≡ k ≡ k ≡ k ≡
16 (mod 21) 3 k ≡ k ≡
10 (mod 28) 1 k ≡
24 (mod 28) 2 k ≡ pk ≡
11 (mod 35) 2 k ≡
18 (mod 35) 3 k ≡
25 (mod 70) 1 k ≡
60 (mod 70) 2 k ≡
32 (mod 105) 1 k ≡
67 (mod 105) 2 k ≡
102 (mod 105) 3 k ≡ k ≡
26 (mod 42) 2 congruence pk ≡
12 (mod 42) 3 k ≡
33 (mod 84) 1 k ≡
75 (mod 84) 2 k ≡
19 (mod 63) 1 k ≡
40 (mod 63) 2 k ≡
61 (mod 63) 3 k ≡ k ≡
13 (mod 56) 1 k ≡
41 (mod 56) 2 d = 1 (Part II) congruence pk ≡
20 (mod 140) 1 k ≡
48 (mod 140) 2 k ≡
76 (mod 140) 3 k ≡
104 (mod 140) 4 congruence pk ≡
132 (mod 140) 5 k ≡
27 (mod 112) 1 k ≡
83 (mod 112) 2 k ≡
55 (mod 168) 1 congruence pk ≡
111 (mod 168) 2 k ≡
167 (mod 168) 3
Table 12: Covering information for d = 3 (Part I) congruence pk ≡ k ≡ k ≡ k ≡ k ≡
10 (mod 24) 1 k ≡
22 (mod 72) 1 k ≡
46 (mod 72) 2 k ≡
70 (mod 72) 3 k ≡ k ≡
12 (mod 13) 1 k ≡
11 (mod 28) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
41 (mod 74) 2 k ≡ k ≡
80 (mod 111) 1 k ≡
44 (mod 111) 2 k ≡ k ≡
83 (mod 222) 1 k ≡
47 (mod 222) 2 k ≡
11 (mod 222) 3 k ≡
197 (mod 222) 4 k ≡
13 (mod 148) 1 k ≡
87 (mod 148) 2 k ≡
125 (mod 148) 3 k ≡
51 (mod 148) 4 k ≡
89 (mod 148) 5 k ≡
15 (mod 148) 6 k ≡
53 (mod 444) 1 k ≡
275 (mod 444) 2 k ≡
17 (mod 444) 3 k ≡
239 (mod 444) 4 k ≡
129 (mod 296) 1 k ≡
203 (mod 296) 2 k ≡
277 (mod 296) 3 k ≡
647 (mod 888) 2 k ≡
833 (mod 888) 1 congruence pk ≡
611 (mod 888) 3 k ≡
389 (mod 888) 4 k ≡
167 (mod 592) 1 k ≡
463 (mod 592) 2 k ≡
353 (mod 1776) 1 k ≡ k ≡ k ≡
797 (mod 1776) 4 k ≡ k ≡
131 (mod 1332) 2 k ≡
575 (mod 1332) 3 k ≡ k ≡
467 (mod 1332) 5 k ≡
317 (mod 333) 1 k ≡
95 (mod 333) 2 k ≡
206 (mod 333) 3 k ≡
281 (mod 333) 4 k ≡
59 (mod 666) 1 k ≡
503 (mod 666) 2 k ≡
245 (mod 666) 3 k ≡
23 (mod 666) 4 k ≡
659 (mod 1332) 6 k ≡
61 (mod 185) 1 k ≡
172 (mod 185) 2 k ≡
98 (mod 185) 3 k ≡
24 (mod 185) 4 k ≡
321 (mod 370) 1 k ≡
247 (mod 370) 2 k ≡
173 (mod 555) 1 k ≡
284 (mod 555) 2 k ≡
26 (mod 555) 3 k ≡
137 (mod 555) 4 k ≡
248 (mod 555) 5 k ≡
359 (mod 555) 6 k ≡
101 (mod 1110) 1 k ≡
767 (mod 1110) 2 k ≡
693 (mod 740) 1 k ≡
323 (mod 740) 2 k ≡
249 (mod 740) 3 k ≡
619 (mod 740) 4 congruence pk ≡
361 (mod 740) 5 k ≡
731 (mod 740) 7 k ≡
657 (mod 740) 6 k ≡
287 (mod 740) 8 k ≡
213 (mod 740) 9 k ≡
583 (mod 740) 10 k ≡
875 (mod 2220) 1 k ≡
395 (mod 2220) 2 k ≡ k ≡ k ≡ k ≡ k ≡
695 (mod 1480) 2 k ≡
955 (mod 1480) 3 k ≡
215 (mod 1480) 4 k ≡
475 (mod 1480) 5 k ≡ k ≡ k ≡ k ≡
509 (mod 4440) 4 k ≡ k ≡ k ≡ k ≡
251 (mod 2960) 3 k ≡
991 (mod 2960) 4 k ≡
547 (mod 2960) 5 k ≡ k ≡ k ≡ k ≡ k ≡
473 (mod 2664) 2 k ≡ k ≡
29 (mod 2664) 4 k ≡ k ≡ k ≡
917 (mod 5328) 3 k ≡ k ≡ k ≡ k ≡ d = 3 (Part II) congruence pk ≡
29 (mod 1665) 1 k ≡ k ≡
584 (mod 1665) 3 k ≡ k ≡ k ≡
881 (mod 3330) 3 k ≡ k ≡
437 (mod 3330) 5 k ≡ k ≡
178 (mod 925) 3 k ≡
733 (mod 925) 2 k ≡
363 (mod 925) 1 k ≡
918 (mod 925) 5 k ≡
548 (mod 925) 4 k ≡
104 (mod 925) 6 k ≡
659 (mod 925) 7 k ≡
289 (mod 925) 8 k ≡
844 (mod 925) 9 k ≡ k ≡
401 (mod 1850) 2 k ≡
31 (mod 1850) 3 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
512 (mod 2775) 5 k ≡ pk ≡ k ≡ k ≡ k ≡ k ≡ k ≡
623 (mod 5550) 4 k ≡
179 (mod 5550) 5 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
217 (mod 259) 1 k ≡
106 (mod 259) 2 k ≡
254 (mod 259) 3 k ≡
143 (mod 259) 4 k ≡
180 (mod 259) 5 k ≡
69 (mod 518) 1 k ≡
329 (mod 518) 2 k ≡
218 (mod 777) 1 k ≡
107 (mod 777) 2 k ≡
773 (mod 777) 3 k ≡
551 (mod 777) 4 k ≡
440 (mod 777) 5 k ≡
959 (mod 1554) 1 k ≡
71 (mod 1554) 3 congruence pk ≡
737 (mod 1554) 2 k ≡ k ≡ k ≡
293 (mod 1554) 6 k ≡
809 (mod 1036) 1 k ≡
921 (mod 1036) 2 k ≡ k ≡
220 (mod 407) 1 k ≡
331 (mod 407) 2 k ≡
35 (mod 407) 3 k ≡
257 (mod 407) 4 k ≡
368 (mod 407) 5 k ≡
72 (mod 407) 6 k ≡
183 (mod 814) 1 k ≡
701 (mod 814) 3 k ≡
405 (mod 814) 2 k ≡
109 (mod 814) 4 k ≡
517 (mod 814) 5 k ≡
221 (mod 1221) 1 k ≡
332 (mod 1221) 2 k ≡
554 (mod 1221) 3 k ≡
665 (mod 1221) 4 k ≡
776 (mod 1221) 5 k ≡
887 (mod 1221) 6 k ≡ k ≡ k ≡ Table 13: Covering information for d = 4 congruence pk ≡ k ≡ k ≡ k ≡ k ≡
14 (mod 18) 1 k ≡ k ≡
17 (mod 27) 1 k ≡ k ≡
26 (mod 54) 1 congruence pk ≡
53 (mod 54) 2 k ≡
15 (mod 81) 1 k ≡
33 (mod 81) 2 k ≡
51 (mod 81) 3 k ≡
69 (mod 81) 4 k ≡ k ≡
105 (mod 162) 1 k ≡
123 (mod 162) 2 k ≡
141 (mod 162) 3 congruence pk ≡
159 (mod 162) 4 k ≡ k ≡
27 (mod 144) 1 k ≡
99 (mod 144) 2 k ≡
63 (mod 216) 1 k ≡
135 (mod 216) 2 k ≡
207 (mod 432) 1 k ≡
423 (mod 432) 2
Table 14: Covering information for d = 6 congruence pk ≡ k ≡ k ≡ k ≡ k ≡
13 (mod 20) 2 k ≡ k ≡
17 (mod 30) 2 congruence pk ≡
27 (mod 30) 3 k ≡ k ≡
28 (mod 40) 1 k ≡
18 (mod 60) 1 k ≡
58 (mod 60) 2 k ≡
38 (mod 60) 3 k ≡ pk ≡ k ≡
29 (mod 45) 1 k ≡
14 (mod 45) 2 k ≡
89 (mod 90) 1 k ≡
44 (mod 90) 2 d = 7 congruence pk ≡
16 (mod 66) 2 k ≡
136 (mod 264) 6 k ≡
202 (mod 264) 7 k ≡
70 (mod 528) 1 k ≡
268 (mod 528) 2 k ≡
334 (mod 528) 3 k ≡
532 (mod 1584) 1 k ≡ k ≡ k ≡ k ≡
11 (mod 32) 1 k ≡
27 (mod 32) 2 k ≡ k ≡ k ≡
10 (mod 16) 1 k ≡ k ≡
13 (mod 22) 2 k ≡ k ≡
14 (mod 44) 1 k ≡
36 (mod 44) 2 k ≡
15 (mod 33) 1 k ≡
26 (mod 33) 2 k ≡
37 (mod 66) 1 k ≡
38 (mod 132) 1 k ≡
126 (mod 132) 2 k ≡
60 (mod 132) 3 k ≡ k ≡
49 (mod 88) 2 k ≡
104 (mod 264) 1 k ≡
236 (mod 264) 2 k ≡
71 (mod 264) 3 k ≡
159 (mod 264) 4 k ≡
247 (mod 264) 5 k ≡ k ≡
17 (mod 55) 2 k ≡
28 (mod 55) 3 k ≡
39 (mod 55) 4 k ≡
50 (mod 110) 1 k ≡
105 (mod 110) 2 k ≡ k ≡
62 (mod 110) 4 k ≡
18 (mod 220) 1 k ≡
73 (mod 220) 2 k ≡
128 (mod 220) 3 k ≡
183 (mod 220) 4 k ≡
40 (mod 275) 1 congruence pk ≡
95 (mod 275) 2 k ≡
150 (mod 275) 3 k ≡
29 (mod 165) 1 k ≡
84 (mod 165) 2 k ≡
139 (mod 165) 3 k ≡
51 (mod 220) 5 k ≡
161 (mod 220) 6 k ≡
106 (mod 330) 1 k ≡
216 (mod 330) 2 k ≡
326 (mod 330) 3 k ≡ k ≡
19 (mod 77) 2 k ≡
30 (mod 77) 3 k ≡
41 (mod 77) 4 k ≡
52 (mod 154) 1 k ≡
129 (mod 154) 2 k ≡
63 (mod 154) 3 k ≡
140 (mod 154) 4 k ≡
74 (mod 154) 5 k ≡
151 (mod 154) 6 k ≡ k ≡
20 (mod 99) 2 k ≡
31 (mod 99) 3 k ≡
42 (mod 99) 4 k ≡
53 (mod 198) 1 k ≡
152 (mod 198) 2 k ≡
64 (mod 396) 1 k ≡
163 (mod 396) 2 k ≡
262 (mod 396) 3 k ≡
361 (mod 792) 1 k ≡
757 (mod 792) 2 k ≡
75 (mod 297) 1 k ≡
174 (mod 297) 2 k ≡
273 (mod 297) 3 k ≡
86 (mod 297) 4 k ≡
185 (mod 297) 5 k ≡
284 (mod 297) 6 k ≡
97 (mod 594) 1 k ≡
196 (mod 594) 2 k ≡
295 (mod 594) 3 k ≡
394 (mod 594) 4 k ≡
493 (mod 594) 5 k ≡
592 (mod 1188) 1 k ≡ k ≡
10 (mod 121) 1 k ≡
21 (mod 121) 2 congruence pk ≡
32 (mod 121) 3 k ≡
43 (mod 121) 4 k ≡
54 (mod 242) 1 k ≡
175 (mod 242) 2 k ≡
65 (mod 242) 3 k ≡
186 (mod 242) 4 k ≡
76 (mod 242) 5 k ≡
197 (mod 484) 1 k ≡
439 (mod 484) 2 k ≡
87 (mod 484) 3 k ≡
208 (mod 484) 4 k ≡
98 (mod 363) 1 k ≡
219 (mod 363) 2 k ≡
109 (mod 726) 1 k ≡
230 (mod 726) 2 k ≡
351 (mod 726) 3 k ≡
593 (mod 1452) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
120 (mod 605) 1 k ≡
362 (mod 1210) 1 k ≡
967 (mod 1210) 2 k ≡ k ≡ k ≡
604 (mod 2420) 1 k ≡ k ≡ k ≡ k ≡ k ≡ k ≡
205 (mod 275) 4 k ≡
260 (mod 275) 5 k ≡
329 (mod 484) 5 k ≡
450 (mod 484) 6 k ≡
340 (mod 363) 3 k ≡
472 (mod 726) 4 k ≡ k ≡
714 (mod 2904) 1 k ≡ k ≡
241 (mod 605) 2 k ≡
483 (mod 1210) 3
Table 16: Covering information for d = 9 congruence pk ≡ k ≡ pk ≡ pk ≡≡