Hexagonal Triangular Squares - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:11:22Z http://mathoverflow.net/feeds/question/31912 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31912/hexagonal-triangular-squares Hexagonal Triangular Squares Thomas P Hayes 2010-07-14T22:06:40Z 2013-02-27T01:08:40Z <p>Is there a hexagonal, triangular, square (apart from 0 and 1)?<br> In other words, is there a positive integer that is simultaneously</p> <p>(1) a perfect square, $n^2$, $n \ge 2$,</p> <p>(2) a triangular number, $\frac{m(m+1)}{2}$, $m$ an integer,</p> <p>and (3) a (centered) hexagonal number, $(p+1)^3 - p^3$, $p$ an integer?</p> http://mathoverflow.net/questions/31912/hexagonal-triangular-squares/31919#31919 Answer by Ale De Luca for Hexagonal Triangular Squares Ale De Luca 2010-07-14T23:06:08Z 2010-07-14T23:06:08Z <p>1225 is not a centered hexagonal number (which is the usual term for (3), see <a href="http://en.wikipedia.org/wiki/Centered_hexagonal_number" rel="nofollow">http://en.wikipedia.org/wiki/Centered_hexagonal_number</a> )</p> http://mathoverflow.net/questions/31912/hexagonal-triangular-squares/31922#31922 Answer by Max Alekseyev for Hexagonal Triangular Squares Max Alekseyev 2010-07-14T23:07:13Z 2013-02-27T01:08:40Z <p>The question is equivalent to the system of quadratic Diophantine equations: $$8 n^2 = m'^2 - 1$$ $$4n^2 = 3p'^2 + 1$$ where $m'=2m+1$ and $p'=2p+1$. How to solve such systems is described in my paper: <a href="http://arxiv.org/abs/1002.1679" rel="nofollow">http://arxiv.org/abs/1002.1679</a> (see Theorem 6)</p> <p>It is easy to obtain that the only nonegative solution is $n=1$, $m=1$, $p=0$.</p> http://mathoverflow.net/questions/31912/hexagonal-triangular-squares/31923#31923 Answer by dke for Hexagonal Triangular Squares dke 2010-07-14T23:08:57Z 2010-07-14T23:08:57Z <p>The only solution is 1 - this was a question asked in a book by Gardner and proved by Charles Grinstead, <i>On a Method of Solving a Class of Diophantine Equations </i>, Mathematics of Computation, Vol. 32, No. 143 (Jul., 1978), pp. 936-940.</p> <p>See <a href="http://www.jstor.org/pss/2006498" rel="nofollow">http://www.jstor.org/pss/2006498</a>. </p> <p>(N.B. The hexagonal numbers you are using are actually the <i>centred</i> hexagonal numbers - it's obvious usual hexagonal numbers are always triangular.)</p>