A number theory problem by Kok Hao

Number Theory Level 3

Let $x$ and $y$ be non-zero integers.

How many sets of solutions $(x,y)$ exist that satisfy the below equation?

$256{ x }^{ 8 }+6304 = { y }^{ 8 }-257$

6 2 0 3 5 4 1

1 solution

Kok Hao
Jan 13, 2018

Solution:

$256{ x }^{ 8 }+6304={ y }^{ 8 }-257\\ 256{ x }^{ 8 }+6561={ y }^{ 8 }\\ { (2x) }^{ 8 }+{ 3 }^{ 8 }={ y }^{ 8 }$

Since $x$ and $y$ are non-zero integers, this will cause a direct contradiction to Fermat's Last Theorem, which states that ${ x }^{ n }+{ y }^{ n }={ z }^{ n }$ has no solutions for $n\ge 3$ .

A number theory problem by Kok Hao

1 solution

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