A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occuring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| ≤ 8000 logN/ log logN for sufficiently large N . The proof combines results from extremal graph theory with numbe...

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers a and b giving

We investigate the problem of finding integers k such that appending any number of copies of the base-ten digit d to k yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits such that repeatedly appending any digit yields a composite number.

A coprime labeling of a simple graph of order n is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the first n positive integers. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around the vertices of the ladder. Furthermore, we discuss c...

We deal with a conjecture of Terai (1994) asserting that if a, b, c are fixed coprime integers with min(a, b, c) > 1 such that a+b = c for a certain odd integer r > 1, then the equation a + b = c has only one solution in positive integers with min(x, y, z) > 1. Co-operation man-machine is needed for the proof.