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 1 of 5: Write

a,b,c

in lowest terms

Previous step Next step

a=\frac{a_1}{b_1},\ b=\frac{a_2}{b_2},\ \gcd(a_1,b_1)=\gcd(a_2,b_2)=1,\quad c=\frac{b_1b_2}{a_1a_2}

Detailed analysis

Write $a=a_1/b_1$ and $b=a_2/b_2$ in lowest terms. Since $abc=1$ , $c=b_1b_2/(a_1a_2)$ . Clearing denominators, the hypothesis that $a^x+b^y+c^z$ is an integer becomes the divisibility $a_1^za_2^zb_1^xb_2^y\mid a_1^{x+z}a_2^zb_2^y+a_1^za_2^{y+z}b_1^x+b_1^{x+z}b_2^{y+z}$ .