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%I #38 Sep 08 2022 08:45:54
%S 1373,1097,853,641,461,313,197,113,61,41,53,97,173,281,421,593,797,
%T 1033,1301,1601,1933,2297,2693,3121,3581,4073,4597,5153,5741,6361,
%U 7013,7697,8413,9161,9941,10753,11597,12473,13381,14321,15293,16297,17333,18401
%N Prime-generating polynomial: 16*n^2 - 292*n + 1373.
%C The polynomial generates 31 primes in row starting from n=0.
%C The polynomial 16*n^2 - 668*n + 7013 generates the same primes in reverse order.
%C Note: all the polynomials of the form p^2*n^2 +- p*n + 41, p^2*n^2 +- 3*p*n + 43, p^2*n^2 +- 5*p*n + 47, ..., p^2*n^2 +- (2k+1)*p*n + q, ..., p^2*n^2 +- 79*p*n + 1601, where q is a (prime) term of the Euler polynomial q = k^2 + k + 41, from k=0 to k=39, have their discriminant equal to -163*p^2; the demonstration is easy: the discriminant is equal to b^2 - 4ac = (2k+1)^2*p^2 - 4*q*p^2 = - p^2 ((2k+1)^2 - 4q) = - p^2*(4k^2 + 4k + 1 - 4k^2 - 4k - 164) = -163*p^2.
%C Observation: many of the polynomials formed this way have the capacity to generate many primes in row. Examples:
%C 9n^2 + 3n + 41 generates 27 primes in row starting from n=0 (and 40 primes for n = n-13);
%C 9n^2 - 237n + 1601 generates 27 primes in row starting from n=0;
%C 16n^2 + 4n + 41 generates, for n = n-21 (that is 16n^2 - 668n + 7013) 31 primes in row.
%H Bruno Berselli, <a href="/A181969/b181969.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Coman, <a href="https://www.researchgate.net/profile/Marius_Coman/publication/277912540">Ten prime-generating quadratic polynomials</a>, Preprint 2015.
%H Factor Database, <a href="http://www.factorization.ath.cx/index.php?query=16*n%5E2+-+292*n+%2B+1373&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=50&format=1">Factorizations of 16n^2-292n+1373</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1373-3022*x+1681*x^2)/(1-x)^3. - _Bruno Berselli_, Apr 06 2012
%t Table[16*n^2 - 292*n + 1373, {n, 0, 50}] (* _T. D. Noe_, Apr 04 2012 *)
%o (Magma) [n^2-73*n+1373: n in [0..172 by 4]]; // _Bruno Berselli_, Apr 06 2012
%o (PARI) a(n)=16*n^2-292*n+1373 \\ _Charles R Greathouse IV_, Jun 17 2017
%K nonn,easy
%O 0,1
%A _Marius Coman_, Apr 04 2012
%E Offset changed from 1 to 0 by _Bruno Berselli_, Apr 06 2012
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