Thread: given positive integers a and b such that a|b^2, b^2|a^3, a^3|b^4,...

The problem below seems fairly easy... but I just can't manage to prove it!

PROBLEM: Given posistive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5, ... , prove that a = b.

Originally Posted by swallenberg

The problem below seems fairly easy... but I just can't manage to prove it!

PROBLEM: Given posistive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5, ... , prove that a = b.

this already tells us that every prime that divides a divides b and every prime that divides b divides a, so a,b are divisible by exactly the same primes. Well, now just prove that if a prime divides a to some power, then at that same power that primes divides b, and the other way around.

Tonio

Originally Posted by tonio

Well, now just prove that if a prime divides a to some power, then at that same power that primes divides b, and the other way around.

Tonio

But how? I came this far when I tried the first time... but I just can't seem to cross the finish line.

quote=swallenberg;446066]But how? I came this far when I tried the first time... but I just can't seem to cross the finish line.[/quote]


Suppose and these powers are maximal in each case, i.e. , and suppose , then:

. But . Continue from here comparing odd powers of a with even powers of b.

Tonio

I finally realized how to do it.

(2n-1)alfa ≤ 2nbeta ≤ (2n+1)alfa, which is the same as (2n-1)/2n ≤ beta/alfa ≤ (2n+1)/2n.

Now if we let n tend to inf. we'll see that 1 ≤ beta/alfa ≤ 1 which means that alfa = beta.