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2015-09-16, 14:39   #2
R.D. Silverman

Nov 2003

26·113 Posts

Quote:
 Originally Posted by fivemack A friend of mine observed that the centred hexagonal numbers (3x^2+3x+1) never looked 'nice'. This is true, and the reason is that they're never divisible by 2, 3, 5 or 11. So I've been looking for polynomials which take only values not divisible by small primes. You can construct these by the Chinese Remainder Theorem, but I'm more interested in the asymptotics of where they turn up in nature for polynomials with small coefficients. For quadratics, going up by minimum-value-of-maximum-coefficient: Code: a2 a1 a0 smallest prime with root mod p 48 42 73 53 81 141 -139 59 178 74 179 67 358 222 331 73 786 912 -811 83 (I have examples with 89 [387 7353 -7963] and 97 [960 330 7951] but am not sure they're smallest) For cubics Code: a3 a2 a1 a0 p_min 12 3 33 -1 37 16 9 36 -1 41 4 48 -10 47 47 1 49 -52 1 53 60 52 -10 71 71 6 44 164 -131 79 For monic quadratics Code: 1 1415 -673 67 1 6663 -1361 71 1 8433 -8171 73 1 12431 -6079 83 1 12787 -3187 101 1 16159 -10093 107 For quartics Code: 2 -4 -2 4 -1 23 4 0 -6 0 1 31 3 -6 -4 7 1 41 5 6 18 15 -1 43 9 13 19 13 -1 53 29 0 -29 0 -1 61 4 22 -45 19 -1 83
One may find a lot of useful information in:

http://websites.math.leidenuniv.nl/a...chebotarev.pdf