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

Abstract

In this paper, we have solved the Diophantine equation ቄ൫ ൯ ቅ + = ࢠ and ቄ൫ ൯ ቅ + = ࢠ where ∈ ା and p is an odd prime. Also, we have discussed the generalization of ሺ ሻ + = ࢠ to ቄ൫ ൯ ቅ + = ࢠ , where ∈ ା , q is any prime number and p is an odd prime number. Some solutions of these Diophantine equations have been obtained.

All the Solutions of the Diophantine Equation $p^3 + q^2 = z^3$

“…It is easily seen that infinitely many solutions exist for the equation p 3 + q 1 = z 3 when p is prime and q is prime/composite. Few such examples are: 2 3 + 19 = 3 3 , 3 3 + 37 = 4 3 , 5 3 + 91 = 6 3 , 7 3 + 386 = 9 3 . In this paper, the equation p 3 + q 2 = z 3 yields quite surprisingly only four solutions in all of which p = 7 and in only one of them q 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 [1,3,4,5,6,9,11,13]. The title equation stems from p x + q y = z 2 .…”