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Old 2021-09-10, 13:23   #11
Dr Sardonicus
 
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Feb 2017
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Quote:
Originally Posted by fivemack View Post
To find Machin-like formulae for pi, I want to find sets of N where N^2+1 has only small prime factors. Tangentially, it would be nice to have a proof that, for example, n=485298 is the largest number where n^2+1 has no prime factor greater than 53.
If n2 + 1 has no prime factor greater than 53, we have n2 - dy2 = -1 where d divides 2*5*13*17*29*37*41*53. We then have to find all y divisible by no primes other than those specified.

It occurred to me that some of the quadratic fields \mathbb{Q}\(\sqrt{d}\), d as above, have fundamental units with norm +1; and for these d, n2 - dy2 = -1 will have no solution. Of the 255 values of d which are (non-empty) products of the eight primes 2, 5, 13, 17, 29, 37, 41, and 53, there are 70 such values of d, given below. Not a huge proportion, but it might help a little:

[34, 205, 221, 377, 410, 689, 1394, 1517, 1537, 1802, 1885, 1961, 3034, 4810, 4930, 5945, 6290, 7685, 10730, 11713, 15170, 19610, 19981, 23426, 25493, 26129, 27898, 30914, 33337, 45305, 56498, 56869, 81770, 99905, 117130, 139490, 141245, 197210, 257890, 282490, 335257, 339677, 433381, 439930, 739297, 747881, 804010, 960466, 1478594, 1638442, 1676285, 2166905, 2401165, 3396770, 4096105, 4833865, 7478810, 9667730, 9722453, 13668170, 17768621, 19444906, 23316290, 30311177, 39637693, 97224530, 125680490, 177686210, 303111770, 396376930]
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