Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to know if anyone can estimate the quantity $\frac{\sqrt{p_{n,4}}}{\max(a_p,b_p)}$ where $p_{n,4}$ is the $n$-th prime such that $p = 4k + 1$ for some integer $k$.

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

What kind of estimate are you looking for? Your quantity is between $1$ and $2$, clearly and should oscillate between the two value, probably uniformly. It is widely believed that there are infinitely many primes of the form $n^2+1$ or $n^2+(n+1)^2$ which should give the extremes of the interval.

The quantity is governed by the solutions modulo $p$ of the congruence $x^2+1 \equiv 0 \pmod p$ and, I believe there is a paper of Friedlander and Iwaniec showing that the root of this equation is uniformly distributed in the interval $[1,p]$ in the natural sense.

share|improve this answer
    
You meant $\sqrt{2}$ not $2$, no? –  Chris Wuthrich Jun 7 '13 at 22:38
    
Thank you. I was looking for the supremum and soon after I posted I realized it is obviously $\sqrt{2}$. –  Mustafa Said Jun 7 '13 at 23:11
    
Yes. $\sqrt{2}$, thanks. –  Felipe Voloch Jun 7 '13 at 23:51
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.