If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing

Abstract. We show that the minimum period modulo p of the Bell exponential integers is (p p −1)/(p −1) for all primes p<102 and several larger p. Our proof of this result requires the prime factorization of these periods. For some primes p the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed. 1.

Abstract. 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 = (p p − 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 is an odd positive integer and m is a positive integer and q = 4m 2 p + 1 is prime, then q divides p m2 p − 1. Then we explain how this theorem influences the probability that q divides Np. 1.

Abstract. 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 =(p p − 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 is an odd positive integer and m is a positive integer and q =4m 2 p + 1 is prime, then q divides p m2 p − 1. Then we explain how this theorem influences the probability that q divides Np. 1.