What's your favorite problem in Number Theory?

KDS · Poincare Conjecture Offline Joined: 26 Jun 2009 Posts: 107

Hello,Mathlinkers.What is your favorite number theory problem?Can you introduce it?

Mine:If $p=4k+1$ is a prime,then $k^k \equiv 1 (mod p)$

**Posted:** Mon Jun 29, 2009 3:21 am

_________________
$\mathfrak{KDS}$

Stephen

**Posted:** Sat Jul 04, 2009 2:00 am

uglysolutions · Poincare Conjecture Offline Joined: 14 Jun 2008 Posts: 131

Mine is quite standard:

Prove that the equation $x^2 + y^2 = 3z^2$ has no positive integer solutions.

Or this one, although I don't know if it is number theory or combinatorics:

Prove that among $n$ integers there are always some of them whose sum is divisible by $n$ .

**Posted:** Sat Jul 18, 2009 12:46 pm

ZetaX

About the ones I posted on the forum. This includes:

-If $p \equiv 1 \mod 4$ is prime, then $\frac{p^p-1}{p-1}$ is not prime ( http://www.artofproblemsolving.com/Forum/viewtopic.php?t=169135 ). And the closely related Gauß&Lucas' formulas ( http://www.artofproblemsolving.com/Forum/viewtopic.php?t=146020 ).

- Clausen van Staudt, that is: describing Bernoulli numbers $\mod 1$ ( http://www.artofproblemsolving.com/Forum/viewtopic.php?t=91510 ).

- Euclidean polynomials to prove Dirichlet's theorem on primes in aritmetic progressions in some cases ( http://www.artofproblemsolving.com/Forum/viewtopic.php?t=100567 )

- Erdös-Ginzburg-Zif (or however they are spelled): from $2n-1$ integers, one can pick $n$ such that their sum is divisible by $n$ ( http://www.artofproblemsolving.com/Forum/viewtopic.php?p=389030 ).

There would be some more if we would speak about theorems instead of problems.