4/3/2008

J. Klueners
Duesseldorf

Cohen-Lenstra heuristics and the negative Pell equation

For a squarefree integer $d$ we ask, if the negative Pell equation $x2-dy2 = -1$ is solvable over the integers. By easy considerations we see that in this case $d>0$ and that all odd prime divisors of $d$ are congruent to 1 modulo 4. Now we call a $d$ special, if it satisfies those two conditions. We are able to prove that for a positive density of special $d$ we can solve the negative Pell equation. Furthermore there is a positive density of special $d$, where the negative Pell equation cannot be solved. This result gives a big support to a conjecture of Stevenhagen who predicts those densities.