“Fermat’s Last Theorem”

A puzzle by H.A. Thurston, from the April 1947 issue of Eureka, the journal of recreational mathematics published at Cambridge University:

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The answer appears in the March 1948 issue: “The statements of 2 and 3 are true, those of 1, 4 and 5 are false.” But no full solution is given. (The Eureka archive is published under a Creative Commons license.)