Problem #16

Fermat's Last Theorem (recently proved by Andrew Wiles) states that for all integers n > 2, the equation x^n + y^n = z^n has no positive integer solutions (i.e. solutions where x, y, and z are all positive integers). However, if the exponents are unequal, the situation can be quite different:

i. Find a positive integer solution to x^2 + y^3 = z^4.

ii. Find one for x^3 + y^4 = z^5.

iii. More generally, find one for x^n + y^(n+1) = z^(n+2).

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