For k ≡ 2 mod 4 the equation x^k + (x+1)^k = y^n (n ≥ 3) has only the solutions x = 0, -1 when 6 ≤ k ≤ 100 or k has an odd prime factor ≡ 3 mod 4.
Bennett, Jordan S
2 Pith papers cite this work. Polarity classification is still indexing.
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math.NT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
The authors extend the modular method to Ax² + By^r = Cz^p using Darmon's framework and Frey hyperelliptic curves, then apply it to study the conjecture 5x² + q^{2n} = y^5.
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Sum of consecutive powers as a perfect power
For k ≡ 2 mod 4 the equation x^k + (x+1)^k = y^n (n ≥ 3) has only the solutions x = 0, -1 when 6 ≤ k ≤ 100 or k has an odd prime factor ≡ 3 mod 4.
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The generalized Fermat equation $Ax^2 + By^r = Cz^p$ and applications
The authors extend the modular method to Ax² + By^r = Cz^p using Darmon's framework and Frey hyperelliptic curves, then apply it to study the conjecture 5x² + q^{2n} = y^5.