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 5 of 5: Conclude

a_1

, hence

a

, is powerful

Previous step Next step

k:=\frac{y+z}{\gcd(z,y+z)}>1,\quad k\mid n\ \forall p\mid a_1\implies a_1=t^k,\ \gcd(t,b_1)=1\implies a=\frac{t^k}{b_1}\text{ is powerful}

Detailed analysis

From $nz=m(y+z)$ with $n,m$ positive integers, dividing through by $\gcd(z,y+z)$ shows $n$ must be divisible by $k:=(y+z)/\gcd(z,y+z)$ , and $k>1$ because $z<y+z$ . Since this holds for every prime $p$ dividing $a_1$ , every exponent in the prime factorization of $a_1$ is divisible by $k$ , so $a_1=t^k$ for some positive integer $t$ . Because $\gcd(a_1,b_1)=1$ , also $\gcd(t,b_1)=1$ , and thus $a=t^k/b_1$ has the required form. By the symmetric argument (with the corresponding variables exchanged), $b$ and $c$ are also powerful.