Is the class number of the quadratic field $x^2=3\cdot2^n+1$ $O(n)$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:44:26Z http://mathoverflow.net/feeds/question/79463 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79463/is-the-class-number-of-the-quadratic-field-x23-cdot2n1-on Is the class number of the quadratic field $x^2=3\cdot2^n+1$ $O(n)$? joro 2011-10-29T13:14:06Z 2011-10-29T13:41:03Z <p>Numerical evidence suggest that the class number of the quadratic field $x^2=3\cdot2^n+1$ is $O(n)$ while the discriminant is $O(2^n)$.</p> <p>Here are the class numbers for $n=6 \dots 110$ computed with pari/gp:</p> <pre><code>1, 2, 1, 2, 1, 4, 1, 3, 2, 4, 1, 1, 1, 2, 3, 1, 2, 1, 2, 2, 2, 4, 48, 1, 3, 2, 1, 4, 2, 2, 1, 8, 4, 4, 4, 5, 1, 2, 6, 35, 3, 80, 25, 2, 4, 24, 4, 4, 12, 8, 2, 1, 24, 12, 2, 8, 8, 1, 8, 4, 13, 104, 4, 2, 1, 8, 4, 8, 400, 2, 4, 1, 1, 4, 2, 2, 2, 4, 4, 2, 10, 80, 2, 16, 2, 16, 2, 372, 4, 32, 4, 46, 8, 6, 8, 12, 6, 1, 4, 4, 4, 4, 8, 4, 12, 4, 8, 8, 4, 72 </code></pre> <p><a href="http://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1984c/art.pdf" rel="nofollow">A Monte Carlo factoring algorithm with linear storage</a> possibly might be used to verify for larger $n$, though I don't have working implementation yet.</p> <p>The motivation is that small class number and the above algorithm might give a divisor of numbers of these form.</p> <p>I get similar results for $x^2=4\left(3\cdot2^n-1\right)$</p> <p>For Mersenne numbers $x^2=2^n-1$ there are large class numbers.</p> <pre><code> 1, 1, 4, 2, 8, 2, 4, 1, 16, 4, 48, 9, 16, 1, 16, 2, 256, 2, 272, 4, 520, 4, 1920, 32, 576, 5, 4352, 16, 12168, 16, 6656, 4, 24480, 16, 20416, 4, 8560, 72, 241920, 140352, 498720, 4, 206592, 2, 1494528, 384, 4771328, 32, 758016, 128, 11758848, 8, 19031040, 6, 7610368, 1, 70529424, 16, 246776832, 80, 51142144, 2, 615757824, 16, 1378201600, 8, 900177920, 8, 4452143904, 128, 9139057152, 8, 8201582592, 24, 4530836992, 32, 83129328224, 4, 6856657920, 16, 182429183488, 128, 617893922304, 15, 260395347968, 32, 1944611808768, 4, 3750374341632, 32 </code></pre>