mersenneforum.org On polynomials without roots modulo small p
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 2015-09-16, 14:09 #1 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 23·3·5·53 Posts On polynomials without roots modulo small p 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 2106 546 3037 89 3840 1290 2089 97 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 1 41569 -19993 127 1 97195 -93463 131 1 139831 -124513 137 1 221601 -204983 149 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 3 0 85 0 1 89 7 -14 -90 97 -1 103 29 0 37 0 107 107 61 -122 -41 102 109 109 For ax^4+bx^2+c Code: 4 0 -6 0 1 31 19 0 19 0 -1 37 21 0 -21 0 -1 41 25 0 -27 0 1 47 29 0 -29 0 -1 61 61 0 53 0 -1 83 3 0 85 0 1 89 29 0 37 0 107 107 19 0 -139 0 -137 113 156 0 -174 0 -131 131 9 0 259 0 1 139 486 0 -6 0 -223 149 Last fiddled with by fivemack on 2015-09-17 at 08:19 Reason: add some quadratic-in-x^2 cases
2015-09-16, 14:39   #2
R.D. Silverman

Nov 2003

164448 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

2015-09-18, 12:54   #3
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by R.D. Silverman One may find a lot of useful information in: http://websites.math.leidenuniv.nl/a...chebotarev.pdf
I should have added (from elementary algebraic number theory) that f(x) will be
divisible by p for some x if

(1) p divides the discriminant and
(2) the polynomial is not constant mod p.

Such primes p ramify.

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