John Brillhart

Note on Representing a Prime as a Sum of Two Squares

An improvement is given to the method of Hermite for finding a and b in p = a2 + b2, where p is a prime 3 1 (mod 4).

Some infinite product identities

In this paper we derive the power series expansions of four infinite products of the form 1I (1Xn) 7 (1 + Xn), nES1 nES2 where the index sets Si and S2 are specified with respect to a modulus (Theorems 1, 3, and 4). We also establish a useful formula for expanding the product of two Jacobi triple products (Theorem 2). Finally, we give nonexistence results for identities of two forms.

Tables of Fibonacci and Lucas factorizations

We list the known prime factors of the Fibonacci numbers Fn for n < 999 and Lucas numbers Ln for n < 500. We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject.

On the factors of certain Mersenne numbers. II

This paper is the second of two papers dealing with the prime factors of Mersenne numbers Mp = 2P — 1 of prime exponent (see Brillhart and Johnson [1]). The 899 new factors given below, which were discovered on the IBM 7090 of the Computing Facility at the University of California at Los Angeles, constitute the remaining factors needed for a complete listing in the literature of all prime factors q < 235 for 103 g p g 257 and q < 2U for 257 < p < 20000 (See Brillhart [2], Karst [4], Kravitz [5], Riesel [6], [7]). In pursuing the present investigation, all factors in the literature were rediscovered, which has thus permitted a thorough checking of the tables cited above. Complete agreement was obtained with the tables of Karst and Kravitz, while errors were found in Riesel [7]. (See [9]. Also see Self ridge [8]). It should be noted that the three small factors omitted in Riesel [7] for p = 1451 and 1459 are given in the table below. Also, it should be mentioned that the factors marked in the following table with an asterisk were discovered earlier by E. Karst. The method of search was essentially the same as in [1]. The complete set of factors produced by the program was tested for correctness and multiplicity by a special program written for that purpose. All factors were found to be correct, and none was found to be multiple, thus adding further weight to the conjecture that no multiple factor of Mp exists for p a prime. The factor limit used by the search program was the first multiple of 23040p = 4« 4>(3-5-7-ll-13) greater than 234 (here <t> is the Euler function). Of the trial divisors > 234 only 18504622999, which is a divisor of Mmm , turned out to be a factor. The running time for each Mp , which is an inverse linear function of p, varied from less than one minute for p R¿ 20000 to approximately fifteen minutes for p = 263. A primary concern in the present investigation was the discovery of all M„, 3300 < p < 5000, which had factors < 234. These Mp, being composite, were of interest in a search for prime Mersenne numbers over the same interval, since they could automatically be excluded from that search. (See Hurwitz [3]). No doubt the following table will be of considerable value to investigators in search of large Mersenne primes, who proceed beyond the present range. The author would like to thank John Selfridge and the other directors of the UCLA Computing Facility for their very generous support of this project.

Note on irreducibility testing

An effective method is developed for deducing the irreducibility of a given polynomial with integer coefficients from a single occurrence of a prime value of that polynomial.

Linear quintuple-product identities

In the first part of this paper, series and product representations of four single-variable triple products T0, T1, T2, T3 and four single-variable quintuple products Q0, Q1, Q2, Q3 are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial Q identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper.

On the mod 2 reciprocation of infinite modular-part products and the parity of certain partition functions

An infinite modular-part (MP) product is defined to be a product of the form Π n∈S (1−x n ), where S={n∈Z + / n≡r 1 ,...,r t (mod m)}. A complete method is developed which determines if a given MP product has an MP reciprocal modulo 2 and finds it if it does. Next, a graph-theoretic interpretation of this method is made from which a streamlined algorithm is derived for deciding whether the given MP product is such a reciprocal. This algorithm is applied to the single- variable Jacobi triple product and the quintuple product

The completion of Euler's factoring formula

In this paper we derive a formula for a nontrivial factorization of an odd, composite integer N that has been expressed in two different ways as mx2 + ny2. This derivation is based on an approach that Euler used in a special case in 1778. We also modify this formula to handle the case when N is expressed in two different ways as mx2 −ny2. This latter factorization, however, may sometimes be trivial.

Using Conic Sections to Factor Integers

Abstract This paper explores the factorization of an odd, composite integer N that has been expressed in two different ways as mx2 ± ny2. The negative case mx2 — ny2 = N turns out to be quite different from the positive case mx2 + ny2 = N because it deals with a hyperbola instead of an ellipse. Of particular interest in the negative case is that Pell-connected representations produce trivial factorizations.

Proving Balanced T 2 and Q 2 Identities Using Modular Forms

Abstract In our paper “Algorithms for Finding and Proving Balanced Q 2 Identities” we pointed out that the proof algorithm in that paper had not been able to prove 1,417 of the 97,306 tentative identities which the papers search algorithm had found. To deal with this problem, we give here a more powerful proof algorithm based on a familiar technique in the theory of modular forms. The first six sections of this paper establish the connection with modular forms. As it happens, a given balanced T 2 identity in x is mapped directly to a modular form identity of weight 1 on a particular congruence subgroup. The map itself consists of simply multiplying each term of the T 2 identity by the same power of x, written xĪ . The proof algorithm, which is then applied to this identity, consists basically of the following steps: Move the terms of the identity onto the left side of the equation and expand this side into a power series up to a predetermined degree L. If the first L + 1 terms of this expansion are zero, then the identity is true. When this algorithm was applied to the 1,417 recalcitrant identities, it showed they were all true, as hoped.