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Primes of the form f(n) = n^6 - 48*n^5 + 908*n^4 - 8603*n^3 + 42796*n^2 - 105410*n + 100823 listed by increasing value of n >= 0.
1

%I #31 Nov 05 2019 05:56:56

%S 100823,30467,5419,89,719,1423,947,149,199,1499,3323,4177,2879,359,

%T 2179,21773,84407,231859,527819,1967023,13443239,19869323,55748639,

%U 75716119,101253923,173883799,285153899,449885327,557975279,686780659,1475059259,1759928369

%N Primes of the form f(n) = n^6 - 48*n^5 + 908*n^4 - 8603*n^3 + 42796*n^2 - 105410*n + 100823 listed by increasing value of n >= 0.

%C This polynomial f(n) generates 19 prime numbers consecutively (for n=0 to n=18). In n^2 + n + 17, substitute n -> n^3 - 24*n^2 + 166*n - 318.

%C The polynomial f(n) generates 10790 primes in the first 100000 values. - _Stefan Steinerberger_, Apr 21 2006

%D Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_232.htm">Puzzle 232. Primes and Cubic polynomials</a>, The Prime Puzzles & Problems Connection.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.

%e f(1) = 1^6 - 48*1^5 + 908*1^4 - 8603*1^3 + 42796*1^2 - 105410*1 + 100823 = 30467, a prime number.

%t Select[Table[n^6-48n^5+908n^4-8603n^3+42796n^2-105410n+100823,{n,0,500}], PrimeQ[ # ]&] (* _Stefan Steinerberger_, Apr 21 2006 *)

%Y Cf. A005846, A117624.

%K easy,nonn,less

%O 1,1

%A _Roger L. Bagula_ and Parviz Afereidoon (afereidoon(AT)gmail.com), Apr 18 2006

%E More terms from _Stefan Steinerberger_, Apr 21 2006