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November 14th, 2006, 01:04 AM  #1 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  4abab is a square
Seen on another forum: Find all positive integers a,b>0 such that 4abab is a perfect square. Enjoy! 
November 14th, 2006, 06:26 AM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7 
After solving this problem, I realized that it should have been placed in the College Number Theory section. Anyhow, too late. I copypaste my solution in Latex below. Since Latex is not installed on this forum, you will have to do with the code. The solution for this problem will be posted along with my archives of 250+ interesting (collegelevel) problems on this site in a short while (probably in a few days). Write $4abab=u^{2}$. Then $a=\frac{u^{2}+b}{4b1}$, meaning that $4b1$ divides $4u^{2}+1$. If such is the case, then $4u^{2}+1$ must have at least one prime divisor which equals $3$ modulo $4$. If we can show that this is absurd, i.e that all prime divisors of $(2u)^{2}+1$ have the form $4k+1$, then we will be done.\\ By Euler's criterion, $1$ is a quadratic residue modulo $p$, where $p$ is an odd prime, if and only if $(1)^{\frac{p1}{2}}=1 \ mod \ p$. This implies that $p$ must equal $3$ modulo $4$. In particular, $(2u)^{2}+1$ has only prime factors of the form $4k+1$, which solves the problem. 
November 14th, 2006, 08:00 AM  #3 
Newbie Joined: Nov 2006 Posts: 1 Thanks: 0 
Euler first noted that this expression, along with others such as 8ab  3a  3b, is never a perfect square for positive a and b. It is also never a triangular number.


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