We use pruned enumeration algorithms to find lattice vectors close to a specific target vector for the prime number lattice. These algorithms generate multiplicative prime number relations modulo N that factorize a given integer N . The algorithm New Enum performs the stages of exhaustive enumeration of close lattice vectors in order of decreasing success rate. For example an integer N ≈ 10 can...

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (pp − 1)/(p− 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p...

Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ 1⁄4 ðnÞ, where r 5 s5 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ 1⁄4 p 1 holds with some prime p, as well as those positive integers n such that the equation ’ðnÞ 1⁄4 f ðmÞ holds with some int...

In this paper, a new factorization algorithm is presented, which finds a prime factor p of an integer n in time (D log n)O(1), if 4p − 1 = Db2 where D and b are integers. Hence this algorithm will factor a number efficiently, if it has a prime factor p such that 4p−1 is a product of a small integer and a square. Such primes should be avoided when we select the RSA secret keys. Some generalizati...

We describe a modification to the well-known large prime variant of the multiple polynomial quadratic sieve factoring algorithm. In practice this leads to a speed-up factor of 2 to 2.5. We discuss several implementation-related aspects, and we include some examples. Our new variation is also of practical importance for the number field sieve factoring algorithm. 1. Factoring with two large prim...

In this paper, we study the problem of factoring an RSA modulus N = pq in polynomial time, when p is a weak prime, that is, p can be expressed as ap = u0 + M1u1 + . . . + Mkuk for some k integers M1, . . . ,Mk and k+2 suitably small parameters a, u0, . . . uk. We further compute a lower bound for the set of weak moduli, that is, moduli made of at least one weak prime, in the interval [2, 2] and...

Based on an extended quantifier elimination procedure for discretely valued fields, we devise algorithms for solving multivariate systems of linear congruences over the integers. This includes determining integer solutions for sets of moduli which are all power of a fixed prime, uniform p-adic integer solutions for parametric prime power moduli, lifting strategies for these uniform p-adic solut...

In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form 2n + p, for some integer n ≥ 0 and some prime p. In 1950, Erdős constructed infinitely many counterexamples to Polignac’s conjecture. In this article, we show that there exist infinitely many positive integers that cannot be written in either of the forms Fn + p or Fn− p, where Fn is a Fibonacci...

We consider the partition function bp(n), which counts the number of partitions of the integer n into distinct parts with no part divisible by the prime p. We prove the following: Let p be a prime greater than 3 and let r be an integer between 1 and p−1, inclusively, such that 24r + 1 is a quadratic nonresidue modulo p. Then, for all nonnegative integers n, bp(pn + r) ≡ 0 (mod 2).