On the diophantine equation x10 ± y10 = z2

AbstractWe show that the equations x10 + y10 = z2 and x10 - y10 = z2 have no nontrivial integral solutions. Previous demonstrations of these results depend on the fact that the equation x5 + y5 = kz5 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 IE1 where the results about x5 + y5 = kz5 are not known to be available. We also show that x4 + 3x2y2 + y4 = 5z 2 and x4 - 50x2y2 + 125y4 = z2 have no nontrivial solutions, whereas x4 - 3x2y2 + y4 = 5z2 has infinitely many