The paper introduces particle swarm optimization as a viable strategy to find numerical solution of Diophantine equation, for which there exists no general method of finding solutions. The proposed methodology uses a population of integer particles. The candidate solutions in the feasible space are optimised to have better positions through particle best and global best positions. The methodolo...

If a, b and n are positive integers with b ≥ a and n ≥ 3, then the equation of the title possesses at most one solution in positive integers x and y, with the possible exceptions of (a, b, n) satisfying b = a + 1, 2 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 347. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometri...

In this paper we discuss a method used to find the smallest nontrivial positive integer solutions to a1 + a 6 2 = b 6 1 + b 6 2 + b 6 3 + b 6 4 + b 6 5. The method, which is an improvement over a simple brute force approach, can be applied to search the solution to similar equations involving sixth, eighth and tenth powers.

This problem is a computational investigation of Pythagoras' Theorem and Fermat's Last Theorem. It involves a branch of number theory known as Diophantine problems after Diophantus of Alexandria, who is thought to have lived around AD 250 and was the first mathematician to write comprehensively about number theory. For non-technical and historical background see the "popular science" book [1] a...

We show how to determine the k-th bit of Chaitin’s algorithmically random real number Ω by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N . We also demonstrate two further examples of Ω i...

This paper is written with two purposes in mind. First, it points out some mistakes made in the paper \Optimal deconvolution lter design based on orthogonal principle", recently published in this journal. Secondly, in order to sort out the reason for those mistakes, the relations between inner{ outer factorization, spectral factorization, whitening lters and Diophantine equations in MMSE lter d...

A perfect power is a positive integer of the form a where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2, to the Diophantine equation a − b = k is finite. This conjecture amounts to saying ...