SOLUTIONS TO A LEBESGUE–NAGELL EQUATION

We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

The diophantine equation x2 + paqb = yq

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

A note on the ternary purely exponential diophantine equation Ax + By = Cz with A + B = C2

Let l,m,r be fixed positive integers such that 2| l, 3lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm2 − 1,(l − r)lm2 + 1} > 30, then the equation (rlm2 − 1)x + ((l − r)lm2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

Perfect powers that are sums of squares in a three term arithmetic progression

We determine primitive solutions to the equation [Formula: see text] for [Formula: see text], making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.