Four numbers

Number Theory Level 3

Four positive integers ( $a, b, c, d$ ) satisfy the following: $ab=cd$ Is it possible, that if $k$ is a positive integer, then $a^k+b^k+c^k+d^k$ is a prime number?

Yes, it's possible. No, it's not possible.

1 solution

Áron Bán-Szabó
Aug 30, 2017

Not possible.

$a^k+b^k+c^k+d^k=a^k+b^k+c^k+\dfrac{a^k+b^k}{c^k}=\dfrac{(a^k+c^k)(b^k+c^k)}{c^k}$ Since $a^k+c^k>c^k$ and $b^k+c^k>c^k$ so after the simplification, the two-factor product's factors are bigger, than 1, so it is a composite number.

Four numbers

1 solution

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