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Old 2015-09-16, 14:39   #2
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by fivemack View Post
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
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