Abstract
We show that the equations x 10 + y 10 = z 2 and x 10 - y 10 = z 2 have no nontrivial integral solutions. Previous demonstrations of these results depend on the fact that the equation x 5 + y 5 = kz 5 has no nontrivial integral solutions for k = 1, 2, and 8, whereas our proofs avoid this. Consequently, our proofs work in the weak fragment of arithmetic IE 1 where the results about x 5 + y 5 = kz 5 are not known to be available. We also show that x 4 + 3 x 2 y 2 + y 4 = 5 z 2 and x 4 - 50 x 2 y 2 + 125 y 4 = z 2 have no nontrivial solutions, whereas x 4 - 3 x 2 y 2 + y 4 = 5 z 2 has infinitely many.
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