Let p, q, r, s be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers (a, b, c, d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a4 + b4 +c4 = d4; all small solutions to a4 + b4 = c4 +d4; and the smallest positive integer that can be written in 5 ways as a sum of two coprim...

Let p, q, r, s be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers (a, b, c, d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a4 + b4 +c4 = d4; all small solutions to a4 + b4 = c4 +d4; and the smallest positive integer that can be written in 5 ways as a sum of two coprim...

where Z denotes the set of integers. For positive integers A > 1, B > 1 we defineMn(A,B) to be the set of (a, b) with A ≤ |a| ≤ 2A, B ≤ |b| ≤ 2B such that f(t) is irreducible and Df is squarefree. It is reasonable to expect that for A, B tending to infinity, Mn(A,B) ∼ cnAB, for some positive constant cn. This is probably very difficult to prove. We will apply the abc conjecture to show that Mn(...

The Golomb space (resp. the Kirch space) is set N of positive integers endowed with topology generated by base consisting arithmetic progressions a+bN0={a+bn:n≥0} where a,b∈N and b a (square-free) number, coprime a. It known that connected (and locally connected). By recent result Banakh, Spirito Turek, has trivial homeomorphism group hence topologically rigid. In this paper we prove topologica...

The alternate Sylvester sums are Tm(a, b) = ∑ n∈NR(−1)n, where a and b are coprime, positive integers, and NR is the Frobenius set associated with a and b. In this note, we study the generating functions, recurrences and explicit expressions of the alternate Sylvester sums. It can be found that the results are closely related to the Bernoulli polynomials, the Euler polynomials, and the (alterna...

Let n ≥ 3. This paper is concerned with the equation a3 + b3 = cn, which we attack using a combination of the modular approach (via Frey curves and Galois representations) with obstructions to the solutions that are of Brauer–Manin type. We shall show that there are no solutions in coprime, non-zero integers a, b, c, for a set of prime exponents n having Dirichlet density 28219 44928 ≈ 0.628, a...

Let r and s be coprime nonzero integers with ∆ = r 2 + 4s = 0. Let α and β be the roots of the quadratic equation x 2 − rx − s = 0, and assume that α/β is not a root of 1. We make the convention that |α| ≥ |β|. Put (u n) n≥0 and (v n) n≥0 for the Lucas sequences of the first and second kind of roots α and β whose general terms are given by

Given positive integers a, b, c and d such that c and d are coprime we show that the primes p ≡ c(mod d) dividing ak + bk for some k ≥ 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne. Mathematics Subject Classification (2001). 11N37, 11R45.

This paper describes new algorithms for computing a modular inverse e−1 mod f given coprime integers e and f . Contrary to previously reported methods, we neither rely on the extended Euclidean algorithm, nor impose conditions on e or f . The main application of our gcd-free technique is the computation of an RSA private key in both standard and CRT modes based on simple modular arithmetic oper...

We solve the equation xa + xb + 1= yq in positive integers x, y, a, b and q with a > b and q > 2 coprime to φ(x). This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and p-adic logarithms, the hypergeometric method of Thue and Siegel applied p-adically, local methods, and the algorithmic resolution o...