aa r X i v : . [ m a t h . N T ] F e b ADDITIVE BASES AND NIVEN NUMBERS
CARLO SANNA † Abstract.
Let g ≥ base- g Niven number if it is divisible by the sum of its base- g digits. Assuming Hooley’s Riemann Hypothesis, weprove that the set of base- g Niven numbers is an additive basis, that is, there exists a positiveinteger C g such that every natural number is the sum of at most C g base- g Niven numbers. Introduction
One of the principal problems of additive number theory is to determine, given a set ofnatural numbers A , if there exists a positive integer k such that every natural number (resp.,every sufficiently large natural number) is the sum of at most k elements of A . In such a case, A is said to be an additive basis (resp., an asymptotic additive basis ) of order k .Probably the most famous result of additive number theory is Lagrange’s theorem, proved byLagrange in 1770, which says that the set of perfect squares is an additive basis of order 4. Moregenerally, Waring’s problem asks whether the set of perfect h -th powers is an additive basis,which was answered in the affirmative by Hilbert in 1909. Furthermore, in 1937, Vinogradovproved that every sufficiently large odd number is the sum of three prime numbers, whichimplies that the set of prime numbers is an asymptotic basis of order 4. For an introductionto these classic results see, e.g., Nathanson’s book [17].Let g ≥ g representations are restricted in certain ways. For example, Cilleruelo,Luca, and Baxter [4], improving a result of Banks [1], proved that, for g ≥
5, the set ofnatural numbers whose base- g representations are palindrome is an additive basis of order 3.Rajasekaran, Shallit, and Smith [18] showed that the same is true for g = 3 , g = 2; and they proved that the binary palindromes are an additive basis of order 4. Moreover,Madhusudan, Nowotka, Rajasekaran, and Shallit [15] proved that the set of natural numberswhose binary representations consist of two identical repeated blocks is an asymptotic basis oforder 4, while Kane, Sanna, and Shallit [14] gave a generalization regarding k repeated blocks.For other results of this kind see also [2, 3].A natural number is said to be a base- g Niven number if it is divisible by the sum of its base- g digits. De Koninck, Doyon, and K´atai [7], and (independently) Mauduit, Pomerance, andS´ark¨ozy [16], proved that the number of base- g Niven numbers not exceeding x is asymptoticto c g x/ log x , as x → + ∞ , where c g > Theorem 1.1.
Let g ≥ be an integer. Assuming Hooley’s Riemann Hypothesis, we havethat the set of base- g Niven numbers is an additive basis.
Hooley’s Riemann Hypothesis for an integer a (HRH( a ) for short) states that, for all square-free positive integers m , the Dedekind zeta function Z K of the number field K := Q ( ζ m , m √ a ),where ζ m is a primitive m -th root of unity, satisfies the Riemann hypothesis, that is, if Mathematics Subject Classification.
Primary: 11B13, Secondary: 11A63.
Key words and phrases. additive basis; Niven number; sum of digits. † C. Sanna is a member of GNSAGA of INdAM and of CrypTO, the group of Cryptography andNumber Theory of Politecnico di Torino. Z K ( s ) = 0 for some s ∈ C , with 0 < Re( s ) <
1, then Re( s ) = 1 /
2. We assumed HRH( g ),where g is an integer depending on g , to use some deep results of Frei, Koymans, and Sofos [12](Theorem 2.7 and Theorem 2.8 below) concerning sums of three prime numbers with prescribedprimitive roots. Except for that, our proof of Theorem 1.1 employs only elementary methods.Finding an unconditional proof of Theorem 1.1 and determining the order of the additivebasis of the set of base- g Niven numbers are two natural problems. We checked that everynatural number not exceeding 10 is the sum of at most two base-10 Niven numbers.A related problem stems from considering the multiplicative analog of Niven numbers. Anatural number is said to be a base- g Zuckerman number if it is divisible by the product of itsbase- g digits. De Koninck and Luca [9] (see also [10] for the correction of a numerical error in[9]), and Sanna [19] gave upper and lower bounds for the number of base- g Zuckerman numbersnot exceeding x . In particular, there are at least x . , and at most x . , base-10 Zuckermannumbers not exceeding x , for every sufficiently large x . A question is whether the set of base- g Zuckerman numbers is an additive basis. We checked that every natural number n = 106 notexceeding 10 is the sum of at most four base-10 Zuckerman numbers.2. Preliminaries
Throughout this section, let g ≥ n , thereare uniquely determined d , . . . , d ℓ ∈ { , . . . , g − } , with d ℓ = 0, such that n = P ℓi =1 d i g i − .We let [ n ] g := d , . . . , d ℓ (a string), s g ( n ) := P ℓi =1 d i , and ℓ g ( n ) := ℓ . Moreover, for two strings a and b , we write a ≤ b if a is a substring of b , and we let a | b be the concatenation of a and b .We begin with two simple lemmas. Lemma 2.1.
Let n and s , . . . , s v be positive integers such that s g ( n ) = s + · · · + s v and s v > ( g − v − . Then there exist positive integers n , . . . , n v such that [ n ] g = [ n ] g | · · · | [ n v ] g and | s g ( n i ) − s i | ≤ ( g − v − for i = 1 , . . . , v .Proof. If v = 1 then the claim follows by picking n := n . Hence, assume that v ≥
2. Let n = P ℓj =1 d j g j − , where d , . . . , d ℓ ∈ { , . . . , g − } , with d ℓ = 0. We construct n , . . . , n v in the following way: n := P ℓ j =1 d j g j − , where ℓ is the minimal integer in [1 , ℓ ] such that P ℓ j =1 d j ≥ s ; then n := P ℓ j = ℓ +1 d j g j − ℓ − , where ℓ is the minimal integer in ( ℓ , ℓ ] such that P ℓ j = ℓ +1 d j ≥ s ; and so on, up to n v − := P ℓ v − j = ℓ v − +1 d j g j − ℓ v − − , where ℓ v − is the minimalinteger in ( ℓ v − , ℓ ] such that P ℓ v − j = ℓ v − +1 d j ≥ s v − ; Finally, n v := P ℓj = ℓ v − +1 d j g j − ℓ v − − . Fromthis construction, it follows that s i ≤ s g ( n i ) ≤ s i + g − v X j = i +1 s j − ( g − i ≤ ℓ X j = ℓ i +1 d j ≤ v X j = i +1 s j , for i = 1 , . . . , v −
1. In fact, the first inequality in (1) and s v ≥ ( g − v −
2) ensure that each ℓ , . . . , ℓ v − is well defined. Moreover, (1) with i = v − s v − ( g − v − ≤ s g ( n v ) ≤ s v .In particular, n v > s v > ( g − v − ℓ i , it followsthat d ℓ i = 0, so that [ n i ] g = d ℓ i − +1 , . . . , d ℓ i , for i = 1 , . . . , v , where ℓ := 0 and ℓ v := ℓ .Consequently, [ n ] g = [ n ] g | · · · | [ n v ] g and the proof is complete. (cid:3) Lemma 2.2.
Let n and n , . . . , n v be positive integers such that [ n ] g = [ n ] g | · · · | [ n v ] g and n i is the sum of t i base- g Niven numbers for i = 1 , . . . , v . Then n is the sum of t + · · · + t v base- g Niven numbers.Proof.
The claim follows easily after noticing that if m is a base- g Niven number then g i m isa base- g Niven number for every integer i ≥ (cid:3) The next theorem is a result of additive combinatorics due to Dias da Silva and Hamidoune [11].For every integer h ≥ A of an additive abelian group, let h ∧ A denote the setof the sums of h pairwise distinct elements of A , that is, h ∧ A := (cid:8)P a ∈A ′ a : A ′ ⊆ A , |A ′ | = h (cid:9) . DDITIVE BASES AND NIVEN NUMBERS 3
Theorem 2.3.
Let h be a positive integer, let p be a prime number, and let A ⊆ F p . Then (cid:12)(cid:12) h ∧ A (cid:12)(cid:12) ≥ min (cid:8) p, h |A| − h + 1 (cid:9) . In particular, if |A| ≥ h ( p −
1) + h then h ∧ A = F p . The next lemma shows that every positive integer whose sum of digits satisfies certainproperties can be written as the sum of a bounded number of Niven numbers. Hereafter, write g = g u , where g ≥ u ≥ Lemma 2.4. If n is a positive integer such that:(i) s g ( n ) = p + h for a prime number p and an integer h ∈ [4 g, g ] ;(ii) g is a primitive root modulo p ; and(iii) s g ( n ) > max (cid:8) g − ℓ g ( n ) , g (cid:9) ;then n is the sum of at most g + 1 base- g Niven numbers.Proof.
Put ℓ := ℓ g ( n ), s := s g ( n ), and write n = P ℓ − i =0 d i g i , where d , . . . , d ℓ − ∈ { , . . . , g − } .Also, let I := (cid:8) i ∈ { , . . . , ℓ − (cid:9) : d i = 0 } and I ′ := { i mod t : i ∈ I} , where t is themultiplicative order of g modulo p . By (ii) and recalling that g = g u , we get that t ≥ ( p − / u > ( p − /g . Hence, |I| = X i ′ ∈ I ′ |{ i ∈ I : i ≡ i ′ mod t }| < |I ′ | (cid:18) ℓt + 1 (cid:19) < |I ′ | (cid:18) gℓp − (cid:19) . Since g modulo p has order t , letting A := { g i mod p : i ∈ I} ⊆ F p we have |A| = |I ′ | > |I| gℓp − + 1 ≥ sg − gℓp − + 1 > sg − gs ( g − p − + 1= ( p − s gs + ( g − p − > p − g − > p − g + 8 g ≥ p − h + h, where we used the inequalities |I| ≥ sg − , ℓ < g − s , s > p − > g ; of which the last threeare consequences of (i) and (iii). Hence, Theorem 2.3 yields that h ∧ A = F p . In particular,there exists J ⊆ I such that |J | = h and P i ∈J g i ≡ n (mod p ). As a consequence, letting m := n − P i ∈J g i , it follows easily that s g ( m ) = s − h = p and m ≡ p ), so that m is abase- g Niven number. Thus n = m + P i ∈J g i is the sum of h + 1 base- g Niven numbers and,recalling that h ≤ g , the proof is complete. (cid:3) For all integers q > r , let S q,r be the set of of positive integers n such that:(S1) s g ( n ) ≡ r (mod q );(S2) for all positive integers m such that [ m ] g ≤ [ n ] g and ℓ g ( m ) ≥
36 log ℓ g ( n ), we have s g ( m ) > g − ℓ g ( m ).Recall that the lower asymptotic density of a set of positive integers A is defined as the limitinfimum of |A ∩ [1 , x ] | /x , as x → + ∞ . Lemma 2.5.
Let q > and r be integers. Then S q,r has positive lower asymptotic density.Proof. Let ℓ > q be an integer and let n be a uniformly distributed random integer in { , . . . , g ℓ − } . Then n = P ℓi =1 d i g i − , where d , . . . , d ℓ are independent uniformly distributed C. SANNA random variables in { , . . . , g − } . On the one hand, we have P := Pr (cid:2) ℓ g ( n ) = ℓ and s g ( n ) ≡ r (mod q ) (cid:3) = Pr " d ℓ = 0 and ℓ X i = 1 d i ≡ r (mod q ) = q − X s = 0 Pr d ℓ = 0 and ℓ X i = ℓ − q +1 d i ≡ r − s (mod q ) · Pr " ℓ − q X i = 1 d i ≡ s (mod q ) ≥ g q q − X s = 0 Pr " ℓ − q X i = 1 d i ≡ s (mod q ) ≥ g q . On the other hand, by Hoeffding’s inequality [13, Theorem 2], we havePr k X j = 1 d i + j ≤ g − k ≤ e − k/ , for all k ∈ { , . . . , ℓ } and i ∈ { , . . . , ℓ − k } . Hence, letting y := 36 log ℓ , we get P := Pr h ∃ m ∈ N s.t. [ m ] g ≤ [ n ] g , ℓ g ( m ) ≥ y, s g ( m ) ≤ g − ℓ g ( m ) i ≤ X y ≤ k ≤ ℓi ∈ { ,...,ℓ − k } Pr k X j = 1 d i + j ≤ g − k ≤ ℓ X k ≥ y e − k/ ≤ ℓ e − y/ − e − / < ℓ → , as ℓ → + ∞ . Therefore, for every sufficiently large x >
0, letting ℓ be the greatest integer suchthat g ℓ ≤ x , we obtain (cid:12)(cid:12) S q,r ∩ [1 , x ] (cid:12)(cid:12) x > (cid:12)(cid:12) S q,r ∩ [1 , g ℓ ) (cid:12)(cid:12) g ℓ +1 ≥ P − P g > g q +1 . Hence, S q,r has positive lower asymptotic density. (cid:3) The next result is an easy consequence of an important theorem of Schnirelmann.
Theorem 2.6.
Let A be a set of positive integers such that ∈ A . If A has positive lowerasymptotic density, then A is an additive basis.Proof. Since 1 ∈ A and lim inf n → + ∞ |A ∩ [1 ,n ] | n >
0, it follows that inf n ≥ |A ∩ [1 ,n ] | n >
0, that is, A haspositive Schnirelmann density . Consequently, by Schnirelmann’s Theorem [17, Theorem 7.7],it follows that A is an additive basis. (Note that in [17] they say that A is a basis of finiteorder if there exists a positive integer k such that every natural number is the sum of exactly k elements of A , and that [17, Theorem 7.7] has to be applied to A ∪ { } .) (cid:3) The following deep results of Frei, Koymans, and Sofos [12, Theorem 1.1 and Theorem 1.7]are crucial to the proof of Theorem 1.1.
Theorem 2.7.
Let a = ( a , a , a ) ∈ Z such that no a i is − or a square. Assuming HRH( a i ) for i = 1 , , , we have X n = p + p + p a i primitive root mod p i , for i = 1 , , . Y i = 1 log p i ∼ A a ( n ) n as n → + ∞ , with an explicit factor A a ( n ) ≥ that satisfies A a ( n ) ≥ A a , whenever A a ( n ) > , where A a > is a constant depending only on a . Theorem 2.8.
Let a = − be a nonsquare integer. Then A ( a,a,a ) ( n ) > if and only if ( n mod 420) ∈ R a , where R a is a nonempty set of residues modulo depending only on a . DDITIVE BASES AND NIVEN NUMBERS 5
As a consequence, we obtain the following:
Corollary 2.9.
Let a = − be a nonsquare integer and let δ ∈ (0 , . Assuming HRH( a ) ,for every sufficiently large natural number n such that ( n mod 420) ∈ R a there exist primenumbers p , p , p > n δ such that n = p + p + p and a is a primitive root modulo each of p , p , p .Proof. The claim follows easily from Theorem 2.7 and Theorem 2.8 by noticing that X n = p + p + p p ≤ n δ Y i = 1 log p i ≤ X p ≤ n δ p ≤ n (log n ) ≤ n δ +1 (log n ) = o ( n ) , as n → + ∞ . (cid:3) Proof of Theorem 1.1
Let g ≥ g = g u , where g ≥ u ≥ g ) holds. Put q := 420 and r := r ′ + 18 g , where r ′ is anyfixed element of R g , and let A := S q,r so that, thanks to Lemma 2.5, A has positive lowerasymptotic density. By Theorem 2.6, we have that A ∪ { } is an additive basis.Now, in order to prove Theorem 1.1, it suffices to show that every sufficiently large (de-pending only on g ) element of A is the sum of a bounded number (depending only on g ) ofbase- g Niven numbers. Let n ∈ A be sufficiently large, and let ℓ := ℓ g ( n ), s := s g ( n ), and s ′ := s − g .Clearly, ℓ → + ∞ as n → + ∞ . In particular, ℓ ≥
36 log ℓ for every sufficiently large n .Hence, from (S2) with m = n , we get that s > g − ℓ and s ′ > g − ℓ − g . Consequently, inwhat follows, we can assume that ℓ, s, s ′ are sufficiently large.By (S1), we have s ′ ≡ s − g ≡ r − g ≡ r ′ (mod q ), so that ( s ′ mod 420) ∈ R g and s ′ is sufficiently large. Hence, by Corollary 2.9, there exist prime numbers p , p , p > √ s ′ suchthat s ′ = p + p + p and g is a primitive root modulo each of p , p , p .As a consequence, s = s + s + s where s i := p i + 6 g for i = 1 , ,
3. Hence, by Lemma 2.1,there exist positive integers n , n , n such that [ n ] g = [ n ] g | [ n ] g | [ n ] g and | s g ( n i ) − s i | ≤ g −
2) for i = 1 , ,
3. In particular, s g ( n i ) = p i + h i for some integer h i ∈ [4 g, g ]. Note that[ n i ] g ≤ [ n ] g and ℓ g ( n i ) ≥ s g ( n i ) g − > p i g − > √ s ′ g − ≥ q g − ℓ − gg − >
36 log ℓ. Therefore, from (S2) it follows that s g ( n i ) > g − ℓ g ( n i ).Thus we have proved that each n i satisfies the hypotheses of Lemma 2.4, and consequentlyeach n i is the sum of at most 8 g + 1 base- g Niven numbers. Then, from Lemma 2.2, it followsthat n is the sum of at most 24 g + 3 base- g Niven numbers.The proof is complete.
Remark . An inspection of the proof of Theorem 1.1, in particular Lemma 2.4, shows that,actually, we proved a stronger result: Assuming HRH( g ), the union of { g i : i = 0 , , . . . } andthe set of base- g Niven numbers m such that p = s g ( m ) is a prime number and g is a primitiveroot modulo p is an additive basis.4. Acknowledgements
The computational resources were provided by hpc@polito ( ).The author thanks Daniele Mastrostefano (University of Warwick) for suggestions that im-proved the paper. C. SANNA
References [1] W. D. Banks,
Every natural number is the sum of forty-nine palindromes , Integers (2016), Paper No.A3, 9.[2] J. Bell, K. Hare, and J. Shallit, When is an automatic set an additive basis? , Proc. Amer. Math. Soc. Ser.B (2018), 50–63.[3] J. P. Bell, T. F. Lidbetter, and J. Shallit, Additive number theory via approximation by regular languages ,Developments in language theory, Lecture Notes in Comput. Sci., vol. 11088, Springer, Cham, 2018, pp. 121–132.[4] J. Cilleruelo, F. Luca, and L. Baxter,
Every positive integer is a sum of three palindromes , Math. Comp. (2018), no. 314, 3023–3055.[5] R. Daileda, J. Jou, R. Lemke-Oliver, E. Rossolimo, and E. Trevi˜no, On the counting function for thegeneralized Niven numbers , J. Th´eor. Nombres Bordeaux (2009), no. 3, 503–515.[6] J.-M. De Koninck and N. Doyon, Large and small gaps between consecutive Niven numbers , J. Integer Seq. (2003), no. 2, Article 03.2.5, 8.[7] J.-M. De Koninck, N. Doyon, and I. K´atai, On the counting function for the Niven numbers , Acta Arith. (2003), no. 3, 265–275.[8] J.-M. De Koninck, N. Doyon, and I. K´atai,
Counting the number of twin Niven numbers , Ramanujan J. (2008), no. 1, 89–105.[9] J.-M. De Koninck and F. Luca, Positive integers divisible by the product of their nonzero digits , Port. Math.(N.S.) (2007), no. 1, 75–85.[10] J.-M. De Koninck and F. Luca, Corrigendum to “Positive integers divisible by the product of their nonzerodigits”, portugaliae math. 64 (2007), 1: 75–85 [ MR2298113] , Port. Math. (2017), no. 2, 169–170.[11] J. A. Dias da Silva and Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory ,Bull. London Math. Soc. (1994), no. 2, 140–146.[12] C. Frei, P. Koymans, and E. Sofos, Vinogradov’s three primes theorem with primes having given primitiveroots , Math. Proc. Cambridge Philos. Soc. (2021), no. 1, 75–110.[13] W. Hoeffding,
Probability inequalities for sums of bounded random variables , J. Amer. Statist. Assoc. (1963), 13–30.[14] D. M. Kane, C. Sanna, and J. Shallit, Waring’s theorem for binary powers , Combinatorica (2019), no. 6,1335–1350.[15] P. Madhusudan, D. Nowotka, A. Rajasekaran, and J. Shallit, Lagrange’s theorem for binary squares , 43rdInternational Symposium on Mathematical Foundations of Computer Science, LIPIcs. Leibniz Int. Proc.Inform., vol. 117, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018, pp. Art. No. 18, 14.[16] C. Mauduit, C. Pomerance, and A. S´ark¨ozy,
On the distribution in residue classes of integers with a fixedsum of digits , Ramanujan J. (2005), no. 1-2, 45–62.[17] M. B. Nathanson, Additive Number Theory: The Classical Bases , Graduate Texts in Mathematics, vol.164, Springer-Verlag, New York, 1996.[18] A. Rajasekaran, J. Shallit, and T. Smith,
Additive number theory via automata theory , Theory Comput.Syst. (2020), no. 3, 542–567.[19] C. Sanna, On numbers divisible by the product of their nonzero base b digits , Quaest. Math. (2020),no. 11, 1563–1571. Politecnico di Torino, Department of Mathematical SciencesCorso Duca degli Abruzzi 24, 10129 Torino, Italy
Email address ::