2019 February 24

In 1971 the proof below was given that $x^3 + 117 y^3 = 5$ has no integer solutions, using some difficult computations in the field $\mathbb Q(\sqrt[3]{117})$ .

In 1973 it was observed that the equation reduces to $x^3 \equiv 5$ mod 9, which has no solutions.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/BE6D6D5BF821E25BB03D93AE3A13C19F/S0008439500057970a.pdf/on_d_j_lewiss_equation_x_3117y_3_5.pdf

Finkelstein, R., & London, H. (1971). On D. J. Lewis’s Equation x^3 + 117y^3 = 5. Canadian Mathematical Bulletin, 14(1), 111-111. doi:10.4153/CMB-1971-020-x

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