MathDB

(a) Let

n

be a positive integer. Prove that there exist distinct positive integers

x, y, z

such that x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}. (b) Let

a, b, c

be positive integers such that

a

and

b

are relatively prime and

c

is relatively prime either to

a

or to

b.

Prove that there exist infinitely many triples

(x, y, z)

of distinct positive integers

x, y, z

such that x^a \plus{} y^b \equal{} z^c.

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