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 minimumvalueofmaximumcoefficient:
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