A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequences.
Let A be a multiplicative subgroup of . Define the k-fold sumset of A to be . We show that for . In addition, we extend a result of Shkredov to show that for .
Let a,b,c be fixed coprime positive integers with mina,b,c > 1, and let m = maxa,b,c. Using the Gel’fond-Baker method, we prove that all positive integer solutions (x,y,z) of the equation satisfy maxx,y,z < 155000(log m)³. Moreover, using that result, we prove that if a,b,c satisfy certain divisibility conditions and m is large enough, then the equation has at most one solution (x,y,z) with minx,y,z > 1.
In the paper we discuss the following type congruences: where is a prime, , , and are various positive integers with , and . Given positive integers and , denote by the set of all primes such that the above congruence holds for every pair of integers . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets and inclusion relations between them for various values and . In particular, we prove that for all , and , and for...
In this note we prove that the equation , , has only finitely many positive integer solutions . Moreover, all solutions satisfy , and .
We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and .
Let , , , be positive integers such that , , is even and is odd. In this paper we prove that if and either or is an odd prime power, then the equation has only the positive integer solution with .