This paper is devoted to the generalized Fermat equation xp + yq = zr , where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infini...

In this paper we give necessary and sufficient trace conditions for an n×n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n, k ≥ 2 are integers. We prove a discriminant criterion for every 2×2 matrix over an order R in an algebraic number field to be a sum of cubes and fourth powers of matrices over R. We also show that i...

the notation ∑[ and ∑∗ denoting, respectively, a sum over squarefree integers, and one over integers coprime with the (implicit) modulus, which is q here. By work of Montgomery-Vaughan and Selberg, it is known that one can take ∆ = Q − 1 +N (see, e.g., [IK, Th. 7.7]). There are a number of derivations of (1) from (2); for one of the earliest, see [M1, Ch. 3]. The most commonly used is probably ...

The order of every finite group G can be expressed as a product of coprime positive integers m1, . . . ,mt such that π(mi) is a connected component of the prime graph of G. The integers m1, . . . ,mt are called the order components of G. Some nonabelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symp...

A Chebyshev knot C(a, b, c, φ) is a knot which has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + φ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ ∈ R. We show that any two-bridge knot is a Chebyshev knot with a = 3 and also with a = 4. For every a, b, c integers (a = 3, 4 and a, b coprime), we describe an algorithm that gives all Cheb...

We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the ...

We estimate exponential sums with the Fermat-like quotients fg(n) = gn−1 − 1 n and hg(n) = gn−1 − 1 P (n) , where g and n are positive integers, n is composite, and P (n) is the largest prime factor of n. Clearly, both fg(n) and hg(n) are integers if n is a Fermat pseudoprime to base g, and this is true for all g coprime to n when for any Carmichael number n. On the other hand, our bounds imply...

Given an arbitrary ordered pair of coprime integers (a, b) we obtain a pair of identities of the Rogers–Ramanujan type. These identities have the same product side as the (first) Andrews–Gordon identity for modulus 2ab ± 1, but an altogether different sum side, based on the representation of (a/b − 1)±1 as a continued fraction. Our proof, which relies on the Burge transform, first establishes a...

In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least q-fold degenerate when the filling factor is p/q, where and are coprime integers. Interestingly, the simplest Hamiltonian respecting this t...

We study the elliptic algebras Qn,k(E,τ) introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form family quadratic parametrized coprime integers n>k≥1, an curve E, point τ∈E. consider compare several different definitions provide proofs various statements about them made Odesskii. For example, we show that Qn,k(E,0), Qn,n−1(E,τ) are polynomial rings on n variables. ...