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

Abstract

Abstract. In this short article, it is established for the title equation: (i) No solutions exist when p = 2. (ii) Exactly two solutions exist when p = 3. In both solutions q is prime. (iii) Exactly two solutions exist for each and every prime p > 3 in which q is composite. Some numerical solutions are also exhibited.

The Class of Diophantine Equations p^4 + q^y = z^4 when y = 1, 2, 3 is Insolvable for all Primes p and q

“…For each value y = 1, 2, 3, the respective equation p 4 + q y = z 4 has no solutions. Proof: We have the set of three equations (a) p 4 + q 1 = z 4 , (b) p 4 + q 2 = z 4 , (c) p 4 + q 3 = z 4 . Each case will be considered separately.…”

“…and the divisors of q 3 are 1, q, q 2 and q 3 . Evidently, none of these divisors can be applied in any way to equation (4). It follows that equation 4is impossible.…”

All the solutions of the Diophantine Equation $p^x + (p+4)^y = z^2$ when p, (p + 4) are Primes and x + y = 2, 3, 4

“…In this paper, the known Diophantine equation p x + q y = z 2 [see 1,5,6,7] is considered when p and q are Cousin Primes i.e.,…”

On Solutions of the Diophantine Equations p^3 + q^3 = z^2 and p^3 - q^3 = z^2 when p, q are Primes

“…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 [1,4,5,6].…”