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 2 of 5: Deduce

a_1^z\mid b_2^{y+z}

Previous step Next step

a_1^z\mid b_1^{x+z}b_2^{y+z},\quad \gcd(a_1,b_1)=1 \implies a_1^z\mid b_2^{y+z}

Detailed analysis

The left side of the divisibility above is divisible by $a_1^z$ , and $a_1^z$ clearly divides the first two terms on the right (each already contains a factor $a_1^z$ ); hence $a_1^z$ divides the third term $b_1^{x+z}b_2^{y+z}$ . Since $\gcd(a_1,b_1)=1$ , this forces $a_1^z\mid b_2^{y+z}$ .