On the Diophantine Equation px + qy = z2

Abstract

It has been shown in [2] that the title equation has infinitely many solutions when p = 2 and also when p = 3. In this article, it is established and demonstrated for each prime p > 3, that the equation has a solution for each and every integer x ≥ 1. We also discuss separately two distinct particular cases of the equation. One is related to the Sophie Germain conjecture, and the other to the Goldbach conjecture.

Solutions of the Diophantine Equation 2x + py = z2 When p is Prime

“…The literature contains a very large number of articles on non-linear such individual equations involving primes and powers of all kinds. Among them are for example [1,5,9,10].…”

On Solutions to the Diophantine Equation p^2 + q^2 = z^4

“…Peker and Cenberci [8] worked on the Diophantine equation ሺ4 ሻ ௫ + ‫‬ ௬ = ‫ݖ‬ ଶ and the obtained solutions are (x,y,z,p) = (1,2,5,3), (2,2,5,3)and (k,1,2 nx +1, 2 nx+1 ), where k is a non-negative integer and p is an odd prime number , ݊ ∈ Z ା . Burshtein [1] discussed the conditions for the solution of Diophantine equation p x + q y = z 2 based on the various values of p and q where p, q both are prime such that p < q and differ by an even value k. Burshtein [2] discussed and found that the Diophantine equation p x + q y = z 2 has infinitely many solutions when p = 2, 3 and also demonstrated that if prime p> 3 than the equation has a solution for each and every integer x≥1. Burshtein [3] discussed all the solutions to an open problem of Chotchaisthit on the Diophantine equation 2 x + p y = z 2 when y = 1 and p = 7, 13, 29, 37, 257.…”

“…Burshtein [3] discussed all the solutions to an open problem of Chotchaisthit on the Diophantine equation 2 x + p y = z 2 when y = 1 and p = 7, 13, 29, 37, 257. Kumar, Gupta and Kishan [6] solved the Diophantine equation 61 x +67 y =z 2 and 67 x +73 y =z 2 and proved that the equations have not any non-negative integer solution. In this study, we discuss the Diophantine equation ሼሺ‫ݍ‬ ଶ ሻ ሽ ௫ + ‫‬ ௬ = ‫ݖ‬ ଶ where q is any prime number and p is an odd prime number.…”

“…Lemma 2.3. The Diophantine equation ሼሺ2 ଶ ሻ ሽ ௫ + ‫‬ ௬ = ‫ݖ‬ ଶ has the solutions (x,y,z,p) = (1,2,5,3),(2,2,5,3) and (k,1,2 nx +1, 2 nx+1 ) where k is a non-negative integer[5].…”

A note on the Diophantine Equation p + q + r = M2 and the Goldbach Conjectures

“…The equation (2) If the binary Goldbach Conjecture is true, it was shown [5] for a particular case that equation 1has infinitely many solutions. Here, it will be shown for each and every fixed prime p ≥ 2 and each and every value M ≥ 3, that equation 2has at least one solution in which q and r are primes.…”