István Pink

On generalized Lebesgue-Ramanujan-Nagell equations

Abstract We give a brief survey on some classical and recent results concerning the generalized Lebesgue-Ramanujan-Nagell equation. Moreover, we solve completely the equation x2 + 11a 17b = yn in nonnegative integer unknowns with n ≧ 3 and gcd(x, y) = 1.

Linear combinations of prime powers in binary recurrence sequences

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of 2, 3 and 2, 3, 5 are explicitly determined.

On the Diophantine equation (x + 1)k + (x + 2)k + ... + (2x)k = yn

Abstract In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T k ( x ) = ( x + 1 ) k + ( x + 2 ) k + . . . + ( 2 x ) k . Further, on combining Bakers method with the explicit solution of polynomial exponential congruences (see e.g. [6] ), we show that for 2 ≤ x ≤ 13 , k ≥ 1 , y ≥ 2 and n ≥ 3 the title equation has no solutions.

On a variant of Pillai's problem

In this paper, we find all integers

Diophantine equations with Appell sequences

c

On the number of solutions of binomial Thue inequalities

having at least two representations as a difference between a Fibonacci number and a Tribonacci number.

Power values of polynomials and binomial Thue-Mahler equations

We consider the Diophantine equation

ON THE DIOPHANTINE EQUATION x 2 + d 2 l + 1 = y n

P_n (x) = g(y)