User quanta - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:04:50Z http://mathoverflow.net/feeds/user/13121 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63444/are-there-three-variable-generalizations-of-ramanujans-theta-function Are there three variable generalizations of Ramanujans theta function? Quanta 2011-04-29T17:08:46Z 2011-07-03T13:02:03Z <p>The Ramanujan theta function $$f(a,b) = \sum_{n \in \mathbb Z} a^{n(n+1)/2} b^{n(n-1)/2}$$ satisfies the following Jacobi triple product identity $f(a,b) = (-a;ab)_\infty (-b;ab)_\infty (ab;ab)_\infty$.</p> <p>Are there any three variable generalizations which also satisfy identities like that?</p> http://mathoverflow.net/questions/56107/fermats-last-theorem-in-the-cyclotomic-integers Fermat's Last Theorem in the cyclotomic integers. Quanta 2011-02-20T21:10:20Z 2011-07-01T15:13:26Z <p>Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.</p> <p>I am looking for non-trivial solutions to the Fermat equation FLT(p) in the cyclotomic integer ring $\mathbb{Z}[\zeta_{p}]$ for irregular primes p or any information about how the solutions must be (as a step toward constructing them).</p> <p>George Lowther pointed out in an <a href="http://math.stackexchange.com/questions/21883/does-fermats-last-theorem-hold-for-cyclotomic-integers-in-mathbbq-zeta-37" rel="nofollow">earlier discussion</a> that by <a href="http://dx.doi.org/10.1070/IM1999v063n05ABEH000262" rel="nofollow">Kolyvagin's criterion</a> any solution in $\mathbb{Z}[\zeta_{37}]$ must be in the second case.</p> http://mathoverflow.net/questions/64936/relation-between-isogeny-conics-and-fermats-method-of-infinite-descent Relation between Isogeny, Conics and Fermat's method of infinite descent Quanta 2011-05-13T21:09:59Z 2011-05-13T22:16:24Z <p>Fermat's proof of FLT(4) is an example of infinite descent as is Euler's (or whoever you attribute it to's) proof of FLT(3). There are similar proofs to Fermat's for Diophantine equations like $x^4 + y^4 = 2z^2$.</p> <p>I have unsuccessfully tried to view these proofs in terms of group homomorphisms on conics and elliptic curves but it is not at all clear whether this is possible.</p> <p>Can we reinterpret these infinite descent proofs geometrically, in terms of curves?</p> http://mathoverflow.net/questions/64905/which-book-would-you-like-to-see-texified/64906#64906 Answer by Quanta for Which book would you like to see "texified"? Quanta 2011-05-13T16:45:18Z 2011-05-13T16:45:18Z <p>Marcus - Number Fields</p> http://mathoverflow.net/questions/60544/a-remark-of-mordell-alluding-to-a-local-global-principle-for-cubic-diophantine-eq A remark of Mordell alluding to a local/global principle for cubic Diophantine equations Quanta 2011-04-04T12:10:20Z 2011-04-04T13:54:35Z <p>In Mordell <em>Diophantine Equations</em> he says:</p> <blockquote> <p>In recent years it has been shown that there seems to be a close connection between the number of solutions of f(x,y) = 0 (mod $p^r$) and the existence of rational solutions f(x,y) = 0.</p> </blockquote> <p>Does anyone know what observation this is referring to? Has it been turned into a theorem?</p> http://mathoverflow.net/questions/59020/where-can-i-find-a-modern-write-up-of-heegners-solution-of-gauss-class-number-1 Where can I find a modern write-up of Heegner's solution of Gauss' class number 1 problem? Quanta 2011-03-21T00:40:54Z 2011-03-21T02:40:08Z <p>In a recent MO question someone mentioned Heegner's solution of the Gauss "class number 1" problem which takes the following form:</p> <ul> <li>When the class number of an imaginary quadratic form is 1 an elliptic curve is defined over $\mathbb{Q}$ and a modular function takes on integer values at certain quadratic irrationalities which leads to a collection of Diophantine equations: The solution of which finishes the theorem.</li> </ul> <p>I sadly can't read Heegner's original work (since I cannot read German) but also I don't think it's necessarily the best thing to read for this proof due to an alleged gap. So if anyone recognizes this proof sketch sketch and knows where I could read this in detail that would be wonderful! Thanks.</p> http://mathoverflow.net/questions/57717/representations-as-a-sum-of-cubes-following-jacobi Representations as a sum of cubes following Jacobi Quanta 2011-03-07T19:08:12Z 2011-03-07T23:08:48Z <p>Jacobi connected the generating function counting the number of representations as a square with elliptic trigonometry and use Fourier series to find the exact congruence condition and formula for counting representations as a sum of three squares <a href="http://www.math.osu.edu/~econrad/Jacobi/sumofsq/sumofsq.html" rel="nofollow">[1]</a>.</p> <p>To be precise it was the theta function $$1 + \sum_{n=1}^\infty 2 q^{n^2}$$ </p> <p>I was wondering if it was possible to use this approach on positive cubes $$\sum_{n=1}^\infty q^{n^3}$$ and integer cubes $$\sum_{n=-\infty \ldots \infty} q^{n^3}$$ since there is no useful algebraic object (like the Gaussian integers, quaternions and such) for cubes.</p> http://mathoverflow.net/questions/56633/simple-bijection-between-reals-and-sets-of-natural-numbers/56635#56635 Answer by Quanta for Simple bijection between reals and sets of natural numbers Quanta 2011-02-25T14:39:54Z 2011-02-25T14:45:23Z <ul> <li>A set of natural numbers can denote a sequence of natural numbers like {1,2,3} denotes 1,1,1 and {2,4,6,26} denotes 2,2,2,20.</li> <li>A sequence of natural numbers denotes a real number in a unique way using continued fraction.</li> </ul> <p>For example N = {1,2,3,4,5,...} denotes the sequence 1,1,1,1,1,... which is the golden ratio.</p> http://mathoverflow.net/questions/65226/the-ramanujan-problems Comment by Quanta Quanta 2011-05-17T19:30:09Z 2011-05-17T19:30:09Z Can anyone give a solution to this problem <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/q327.htm" rel="nofollow">imsc.res.in/~rao/ramanujan/collectedpapers/&hellip;</a> please? http://mathoverflow.net/questions/64936/relation-between-isogeny-conics-and-fermats-method-of-infinite-descent/64945#64945 Comment by Quanta Quanta 2011-05-14T17:58:19Z 2011-05-14T17:58:19Z This is the sort of thing I was trying to get at, but asked badly. http://mathoverflow.net/questions/64936/relation-between-isogeny-conics-and-fermats-method-of-infinite-descent Comment by Quanta Quanta 2011-05-14T01:26:02Z 2011-05-14T01:26:02Z In the case of FLT(4), I was thinking of considering the conic $X^2 + Y^2 - 1$ and there is a subset of square points on it, &quot;descent&quot; would prove that the only square points are the trivial solutions. http://mathoverflow.net/questions/61632/what-is-the-high-concept-explanation-on-why-real-numbers-are-useful-in-number-the/61687#61687 Comment by Quanta Quanta 2011-04-14T19:42:49Z 2011-04-14T19:42:49Z @Charles, I can make my statement precise if you wish.. http://mathoverflow.net/questions/61632/what-is-the-high-concept-explanation-on-why-real-numbers-are-useful-in-number-the Comment by Quanta Quanta 2011-04-14T10:05:43Z 2011-04-14T10:05:43Z Can you please give some examples of real numbers being useful in number theory? Hopefully not algebraic numbers because they can be constructed in a finite way unlike the other trancendental real numbers.. http://mathoverflow.net/questions/60544/a-remark-of-mordell-alluding-to-a-local-global-principle-for-cubic-diophantine-eq/60549#60549 Comment by Quanta Quanta 2011-04-04T13:43:13Z 2011-04-04T13:43:13Z Thanks! very interesting http://mathoverflow.net/questions/56107/fermats-last-theorem-in-the-cyclotomic-integers Comment by Quanta Quanta 2011-02-21T00:08:00Z 2011-02-21T00:08:00Z Thank you George, that is what I meant to ask!