On the Diophantine equation cy^l=(x^p-1)/(x-1)
Mohammad Sadek
TL;DR
This paper investigates specific conditions under which the Diophantine equation cy^l=(x^p-1)/(x-1) has no integer solutions, focusing on primes p, c and exponent l.
Contribution
It provides new sufficient conditions that guarantee the non-existence of integer solutions for the given Diophantine equation.
Findings
Identifies conditions preventing solutions when p and c are distinct odd primes.
Establishes bounds on l for which solutions cannot exist.
Contributes to the understanding of the equation's solvability based on prime parameters.
Abstract
Let p, c be distinct odd primes, and l \ge 2 an integer. We find sufficient conditions for the Diophantine equation cy^l=(x^p-1)/(x-1) not to have integer solutions
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Mathematical Dynamics and Fractals