Problem 4

Call a rational number

r

powerful if

r

can be expressed in the form

\dfrac{p^k}{q}

for some relatively prime positive integers

p,q

and some integer

k>1

. Let

a,b,c

be positive rational numbers such that

abc=1

. Suppose there exist positive integers

x,y,z

such that

a^x+b^y+c^z

is an integer. Prove that

a,b,c

are all powerful.

Step 3 of 5: Valuation comparison at a prime

p\mid a_1

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p^n\Vert a_1,\ p^m\Vert b_2,\quad nz\le m(y+z),\quad p^{nz}\Vert a_1^za_2^{y+z}b_1^x,\ p^{m(y+z)}\Vert b_1^{x+z}b_2^{y+z}

Detailed analysis

Fix a prime $p\mid a_1$ and let $p^n\Vert a_1$ , $p^m\Vert b_2$ (so $n,m\ge1$ ). From $a_1^z\mid b_2^{y+z}$ we get $nz\le m(y+z)$ . Since $\gcd(a_1,b_1)=\gcd(a_2,b_2)=1$ , $p$ divides neither $b_1$ nor $a_2$ , so $p^{nz}\Vert a_1^za_2^{y+z}b_1^x$ exactly while $p^{m(y+z)}\Vert b_1^{x+z}b_2^{y+z}$ exactly.