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In the first part of this paper, series and product representations of four single-variable triple products T 0 T_0 , T 1 T_1 , T 2 T_2 , T 3 T_3 and four single-variable quintuple products Q 0 Q_0 , Q 1 Q_1 , Q 2 Q_2 , Q 3 Q_3 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 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.

The groundwork for a theory of quadratic identities involving the classical triple and quintuple products is layed. The approach is through the study and use of affine maps that act on indexing lattices associated with the terms (double sums) in the given identity. The terms of the identity are found to be connected by the invariant of a ternary quadratic form.

An infinite, modular-part (MP) product is defined to be a product of the form Π n ∈ S ( 1 − x n ) {\Pi _{n \in S}}(1 - {x^n}) , where S = { n ∈ Z + : n ≡ r 1 , … , r t ( mod m ) } S = \{ {n \in {{\mathbf {Z}}^ + }:n \equiv {r_1}, \ldots ,{r_t}\;\pmod m} \} . Some products of this kind have a mod 2 \bmod \, 2 reciprocal that is also an MP product, while others do not. A complete method is first 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 then applied to the single-variable Jacobi triple product and the quintuple product to determine the cases when these products have an MP reciprocal ( mod 2 ) \pmod 2 . When this occurs—and this occurs in infinitely many cases—the parity of the associated partition function can readily be found. A discussion is also made of the probability that a given MP product with modulus m has an MP reciprocal ( mod 2 ) \pmod 2 .

We prove the congruence \[ ∏ n = 1 n ≢ 7 ( mod 14 ) ∞ ( 1 − x n ) ≡ ∑ − ∞ ∞ ( x n ( 3 n + 2 ) + x 7 n ( 3 n + 2 ) + 2 ) ( mod 2 ) \prod \limits _{\begin {array}{*{20}{c}} {n = 1} \\ {n \nequiv 7\;\pmod {14}} \\ \end {array} }^\infty {(1 - {x^n}) \equiv \sum \limits _{ - \infty }^\infty {({x^{n(3n + 2)}} + {x^{7n(3n + 2) + 2}})\;\pmod 2} } \] by first establishing a related equation, which reduces to the congruence modulo 2. The method of proof (called "expanding zero") is based on a formula of the authors for expanding the product of two triple products. A second proof of the result more fully explicates the various aspects of the method. A parity result for an associated partition function is also included.

A collection of theorems is developed for testing a given integer N for primality. The first type of theorem considered is based on the converse of Fermat’s theorem and uses factors of N − 1 N - 1 . The second type is based on divisibility properties of Lucas sequences and uses factors of N + 1 N + 1 . The third type uses factors of both N − 1 N - 1 and N + 1 N + 1 and provides a more effective, yet more complicated, primality test. The search bound for factors of N ± 1 N \pm 1 and properties of the hyperbola N = x 2 − y 2 N = {x^2} - {y^2} are utilized in the theory for the first time. A collection of 133 new complete factorizations of 2 m ± 1 {2^m} \pm 1 and associated numbers is included, along with two status lists: one for the complete factorizations of 2 m ± 1 {2^m} \pm 1 ; the other for the original Mersenne numbers.

In this paper we give a collection of parity results for partition functions of the form \[ ∏ n ∈ S ( 1 − x n ) − 1 ≡ ∑ − ∞ ∞ x e ( n ) ( mod 2 ) \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty x^{e(n)} \pmod 2 \] and \[ ∏ n ∈ S ( 1 − x n ) − 1 ≡ ∑ − ∞ ∞ ( x e ( n ) + x f ( n ) ) ( mod 2 ) \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty (x^{e(n)} + x^{f(n)})\pmod 2 \] for various sets of positive integers S, which are specified with respect to a modulus, and quadratic polynomials e ( n ) e(n) and f ( n ) f(n) . Several identities similar to theta function identities, such as \[ ∏ n = 1 n ≢ ± ( 4 , 6 , 8 , 10 ) ( mod 32 ) ∞ ( 1 − x n ) = 1 + ∑ n = 1 ∞ ( − 1 ) n ( x n 2 + x 2 n 2 ) , \prod _{\substack {n = 1\\n \nequiv \pm (4,6,8,10)\pmod {32}}}^\infty (1 - x^n) = 1 + \sum _{n = 1}^\infty (-1)^n (x^{n^2} + x^{2 n^2}), \] and its associated congruence \[ ∏ n = 1 n ≢ 0 , ± 2 , ± 12 , ± 14 , 16 ( mod 32 ) ∞ ( 1 − x n ) − 1 ≡ 1 + ∑ n = 1 ∞ ( x n 2 + x 2 n 2 ) ( mod 2 ) , \prod _{\substack {n = 1\\n \nequiv 0, \pm 2, \pm 12, \pm 14,16 \pmod {32}}}^\infty (1 - x^n)^{-1} \equiv 1 + \sum _{n = 1}^\infty (x^{n^2} + x^{2 n^2}) \pmod 2, \] are also proved.

Four new Ramanujan pairs { a i } \{ {a_i}\} , { b j } \{ {b_j}\} are given along with the theorem that no such pairs exist with a 1 = 1 {a_1} = 1 and a 2 = s {a_2} = s for any s ⩾ 5 s \geqslant 5 . All finite Ramanujan pairs are determined and their significance in bounding the local branching degree in the search tree for such pairs is discussed. The search techniques and programs that were used are also described. The parity of the coefficients in the power series is determined in two of the new identities. Partition interpretations of the six recent identities are also given.

The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is demonstrated by the factorization of the seventh Fermat number F 7 {F_7} and other large numbers of interest.

In this paper we derive the power series expansions of four infinite products of the form \[ ∏ n ∈ S 1 ( 1 − x n ) ∏ n ∈ S 2 ( 1 + x n ) , \prod \limits _{n \in {S_1}} {(1 - {x^n})\;\prod \limits _{n \in {S_2}} {(1 + {x^n}),} } \] where the index sets S 1 {S_1} and S 2 {S_2} 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.

A new prime factor is given for each of the Fermat numbers F12 and F13 (none was previously known for F13). The factoring method used and its machine implementation are discussed. A short table of factors and a current status list are also included. In recent years various investigators have used computers to search for prime factors of the Fermat numbers Fm = 22m + 1, m > 7 (see Selfridge [10], Robinson [7] -[9] , Riesel [6] , Brillhart [1], Wrathall [12] , Morrison and Brillhart [3], [4] ). In our investigation we have found two new factors, namely: 190274191361 = 11613415 * 214 + 1 and 2710954639361 = 41365885 . 216 + 1, which divide F1 2 and F13, respectively. Previously, three prime factors of F1 2 hac been discovered, while F1 3 was only known to be composite (see Paxson [5] ). It is well known that any prime factor of Fm has the form k 2m + 2 + 1, m > 2. In searching for such a factor, we can try dividing Fm by each dk of this form for k less than some search limit Lm. Many composite dk can, of course, be eliminated as trial divisors in advance by sieving on the arithmetic sequence {dk} with small, odd primes (in our program the odd primes less than 500 were used). To discover whether a dk which has survived the sieving is a factor of Fm, we calculate Fm(mod dk) by the usual powering method-here only a sequence of squarings and reductions. Since the residues ri of the powers 22'(mod dk), i m, then this factor can only be discovered during the search for factors of Fn. For example, the factor of F1 3 was found during the investigation of F14, since in this case n m = 1. This procedure was coded in COMPASS assembly language for the CDC 6400 at Received May 20, 1974. AMS (MOS) subject classifications (1970). Primary 10A25, 10A40; Secondary 10-04.

We list the known prime factors of the Fibonacci numbers F n {F_n} for n ≤ 999 n \leq 999 and Lucas numbers L n {L_n} for n ≤ 500 n \leq 500 . We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject.

1. P. T. BATEMAN & R. A. HORN, "A heuristic asymptotic formula concerning the distribution of prime numbers," Math. Comp., v. 16, 1962, p. 363-367. 2. S. CHOWLA, "The representation of a number as a sum of four squares and a prime, " Acta Arith., v. 1, 1935, p. 115-122. 3. G. H. HARDY & J. E. LITTLEWOOD, "Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes," Acta Math., v. 44, 1923, p. 1-70. 4. Ju. V. LINNIK, "An asymptotic formula in an additive problem of Hardy-Littlewood," (Russian), Izv. Akad. Nauk SSSR. Ser. Mat., v. 24, 1960, p. 629-706.

Using the theory of elliptic curves, we show that the class number h (− p ) of the field Q ( −p ) appears in the count of certain factors of the Legendre polynomials P m (x) ( mod p) , where p is a prime >3 and m has the form ( p − e )/ k , with k =2,3 or 4 and p≡e ( mod k) . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2 + αxy + y = x 3 and find an elementary expression for the supersingular polynomial ss p ( x ) whose roots are the supersingular j -invariants of elliptic curves in characteristic p . As a corollary we show that the class number h (− p ) also shows up in the factorization ( mod p) of certain Jacobi polynomials.

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.

An improvement is given to the method of Hermite for finding a a and b b in p = a 2 + b 2 p = {a^2} + {b^2} , where p p is a prime ≡ 1 ( mod 4 ) {\text {prime}} \equiv 1\pmod 4 .