On some Diophantine equations of the form hX 4 + Y 4 = Z n

摘要

We show the impossibility of primitive non-zero solutions to the equation hx4 + y 4 = z n if h = 43 and n ∈ {2, 3, 4, 5, 6} and if h ∈ {3, 19, 67} and n ∈ {3, 4, 5, 6} and finally if h ∈ {11, 163} and n ∈ {4, 5}. The proofs are based mostly on elementary methods with no use of computers and the elliptic curve machinery with the exception of a few cases where the rank determinations of certain elliptic curves were accomplished.