Abstract
In this study, we consider the Diophantine equation xa + ya = pkzb where p is a prime number, gcd(a, b) = 1 and k,a,b∈Z+. We solve this equation parametrically by considering different cases of x and y and find that there exist infinitely many nontrivial integer solutions, where the formulated parametric solutions solve xa + ya = pkzb completely for the case of x = y, x = −y, and either x or y is zero (not both zero). For the case of |x| ≠ |y| and both x and y nonzero, not every solution (x,y,z) is in the parametric forms proposed in Theorem 5, although any (x,y,z) in these parametric forms solves the Diophantine equation.
Highlights
IntroductionIn 2016, Wong and Kamarulhaili solved the Diophantine equation x4 + y4 = pkz (where p is a prime and k is a positive integer) nontrivially in the case of x = y, motivated by incomplete parametric solutions proposed by Ismail (2011) for her Diophantine equation of similar form, x4 + y4 = pkz (where p is a prime, p ∈[2,13] and k is a positive integer)
It is known that there exists no nonzero integer solution to Fermat’s equation xn + yn = zn where n > 2, as proven by Andrew Wiles in 1995 (Andreescu and Andrica, 2002)
In 2016, Wong and Kamarulhaili solved the Diophantine equation x4 + y4 = pkz7 nontrivially in the case of x = y, motivated by incomplete parametric solutions proposed by Ismail (2011) for her Diophantine equation of similar form, x4 + y4 = pkz3
Summary
In 2016, Wong and Kamarulhaili solved the Diophantine equation x4 + y4 = pkz (where p is a prime and k is a positive integer) nontrivially in the case of x = y, motivated by incomplete parametric solutions proposed by Ismail (2011) for her Diophantine equation of similar form, x4 + y4 = pkz (where p is a prime, p ∈[2,13] and k is a positive integer) This is due to her assumption in her proofs that z must always contain the prime p when represented as product of primes (Ismail, 2011).
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