Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2 (x) + b (y) = c (z) is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2 (r) - 1, r + ...

In this paper, we find all the solutions of the Diophantine equation x + 2 513 = y in nonnegative integers x, y, α, β, γ, n ≥ 3 with x and y coprime. In fact, for n = 3, 4, 6, 8, 12, we transform the above equation into several elliptic equations written in cubic or quartic models for which we determine all their {2, 5, 13}-integer points. For n ≥ 5, we apply a method that uses primitive diviso...

In number theory, the prime number theory describes the asymptotic distribution of prime numbers. We all know that there are infinitely many primes,but how are they distributed? Dirichlet’s theorem states that for any two positive coprime integers a and d, there are infinitely many primes which are congruent to a modulo d. A stronger form of Dirichlet’s theorem states that the sum of the recipr...

Let p1, p2,..., pn be a sequence of n pairwise coprime positive integers, and let P = p1p2...pn. Let 0, 1, ...,m − 1 be a sequence of m different colors. Let A be an n ×mP matrix of colors in which row i consists of blocks of pi consecutive entries of the same color, with colors 0 through m − 1 repeated cyclically. The Monochromatic Column problem is to determine the number of columns of A in w...

A graph G with vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1, 2, . . . , |V | such that for every edge xy in E, the labels assigned to x and y are relatively prime or coprime. A graph is called prime if it has a prime labeling. In this paper, we show that generalized Petersen graphs P (n, 3) are not prime for odd n, prime for even n ≤ 100 ...

In this paper we will give a proof of the Thue-Siegel-Roth Theorem, which states that for any algebraic number α and any ǫ > 0 there exists only a finite number of pairs of coprime integers p, q such that ∣ α − p q ∣ ∣ < 1 q2+ǫ . We will follow the proof as it is presented Leveque’s book, [8, ch 4]. This proof also deals with the more general case when p q is allowed to be an algebraic number i...

Erdös conjectured that the Diophantine equation x + y = z has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers n for which p | n implies p3 | n. This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers x, y and z none of which is a perfect cube. This is now demonst...

Given a ∈ Z \ {±1, 0}, we consider the problem of enumerating the integers m coprime to a such that the order of a modulo m is square free. This question is raised in analogy to a result recently proved jointly with F. Saidak and I. Shparlinski where square free values of the Carmichael function are studied. The technique is the one of Hooley that uses the Chebotarev Density Theorem to enumerat...

We prove that the equation A+B = C has no solutions in coprime positive integers when p ≥ 211. The main step is to show that, for all sufficiently large primes p, every Q-curve over an imaginary quadratic field K with a prime of bad reduction greater than 6 has a surjective mod p Galois representation. The bound on p depends on K and the degree of the isogeny between E and its Galois conjugate,...

1. We say that g(n) is additive if g(mn)=g(m)+g(n) holds for every coprime pairs m, n of positive integers . If, moreover, g(p') =g(p)' for every prime power pa , then g(n) is called strongly additive . By p, pi , p2 , . . ., q, qr , q2 , . . . we denote prime numbers, c, Cl, c2i . . . are suitable positive constants . F(n) and x(n) denote the largest and the smallest prime factor of n . The sy...