asymptotic families of ramanujan near-integers? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T15:17:43Z http://mathoverflow.net/feeds/question/8659 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8659/asymptotic-families-of-ramanujan-near-integers asymptotic families of ramanujan near-integers? fellow 2009-12-12T07:17:33Z 2009-12-12T09:08:36Z <p>This is a follow-up to the question on the Ramanujan constant.</p> <p>Can one find an asymptotic family of shockingly near-integers obtained by raising e to some simple algebraic number, in the spirit of Ramanujan's $|\exp(\pi \sqrt{163}) - \text{ some integer }| \le 10^{-12}$? Single numerical coincidences don't impress me (and for good reason too ...).</p> <p>Is there a sequence of such near-integers, whose nearness to integers is closer than what you know exists by pigeonholing exp(alpha) for all alpha of bounded "height" or "complexity"?</p> http://mathoverflow.net/questions/8659/asymptotic-families-of-ramanujan-near-integers/8662#8662 Answer by S. Carnahan for asymptotic families of ramanujan near-integers? S. Carnahan 2009-12-12T08:52:38Z 2009-12-12T09:08:36Z <p>The Ramanujan behavior is typically explained by the fact that the imaginary quadratic field $\mathbb{Q}(\sqrt{-163})$ has class number one (together with an integrality property of the j-function - see <a href="http://en.wikipedia.org/wiki/Heegner%5Fnumber#Almost%5Fintegers%5Fand%5FRamanujan.27s%5Fconstant" rel="nofollow">Wikipedia</a>). Since there are only finitely many such imaginary quadratic fields, you can't really expect to have infinitely many similar phenomena (at least admitting a similar explanation). </p>