Yes, you are exactly right. For those who did not click on the link, the near equality was: 3^3 = (2^2 * 13^2 * 37^2 * 73^2) / (3^2 * 5^2 * 17^2 * 53^2). Which numerically is: 27 = 27.00000000547. I encountered this while looking this morning for consecutive integer sided triangles, sides n, n+1, n+2 with integer area. For n = 140451, the area is: 2*3^3*5*13*17*37*53*73. Since the triangle is nearly equilateral one can get a good rational approximation to sqrt(3) from this, which I modified as above to make it look cute. I then discovered that the sequence of such triangles is well known, is in OEIS as A003500 and corresponds to the Pellian curve x^2-27y*2=1 that you mention. Of course to get "cute" near-equalities you need to find rational approximations to irrationals that involve many small primes and are (nearly) square free. On Wed, Oct 8, 2008 at 1:01 PM, Dave Blackston <hyperdex@gmail.com> wrote:
On Wed, Oct 8, 2008 at 7:45 AM, James Buddenhagen <jbuddenh@gmail.com>wrote:
Most rational approximations are to irrationals. This one, coming from a Heron triangle, is to an integer. Click here: http://i35.tinypic.com/35cmnia.png for a less than 1k formatted version.
Couldn't any number of approximations of this form be obtained by simply taking rational approximations to square roots and then squaring both sides? In particular, this approximation follows after noting that (70226, 13515) solves the Pellian x^2-27y*2=1.