On the equation 𝑠²+𝑦^{2𝑝}=𝛼³

Chen, Imin

doi:10.1090/S0025-5718-07-02083-2

On the equation $s^2+y^{2p} = \alpha ^3$
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Math. Comp. 77 (2008), 1223-1227 Request permission

Abstract:

We describe a criterion for showing that the equation $s^2+y^{2p} = \alpha ^3$ has no non-trivial proper integer solutions for specific primes $p > 7$. This equation is a special case of the generalized Fermat equation $x^p + y^q + z^r = 0$. The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific $p$ in the final stage of the method (1998). The criterion can be computationally verified for primes $7<p < 10^7$ and $p \not = 31$.

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