A280717 Given a prime number p, let b = -p and c = p^2. Assuming that the polynomial P(x) := x^2+b*x+c takes at least one prime value for some positive integer x

3, 7, 43, 1693, 2864557, 8205572225569, 67331415548799635795058613, 4533519519805137360312930667312809111343819483374997, 20552799236454203238557860425684304712780972342513397945121797314302926172950212696842909492430773376197

Offset: 1

Author

Luis H. Gallardo, Jan 07 2017

Keywords

• nonn
• hard

Comments

The next term is only defined if the set S above is not empty.

Conjecture: the sequence is well defined.

a(13) has 1654 digits. If S is not empty, then its maximal element is P(x) where x is the least positive integer x <= p/2 such that P(x) is prime. - Chai Wah Wu, Jan 09 2017

Examples

			a(2) = 7, since 7 = max S_3, where S_3 = {x^2-3x+9 : x is an integer with 0<x<2, and x^2-3x+9 is a prime number}. Clearly, S_3={7}, thus a(2)=7. Now we explain why a(3)=43. We have 43 = max S_7. S_7 := {x^2-7x+49 : x is an integer, 0 <x<7, and x^2-7x+49 is a prime number}. By computations S_7 = {37,43}. Thus a(3) = max S_7 = 43. We explain also why a(4) = 1693. One has 1693 = max S_43, where S_43 = {x^2-43x+43^2 : x is an integer, 0 <x < 43, and x^2-43x+43^2 is a prime number}. By computations S_43 = {1399,1429,1459,1543,1597,1627,1693}. Thus a(3) = max S_43 = 1693.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..12

Programs

Maple

with(numtheory):
xa := proc(aa) local P,x,a,a2,mi,mm; a:= aa; a2 := a^2; mi := 0; for x from 1 to a-1 do P := x^2-a*x+a2; if isprime(P) then mi := max(P,mi); fi; od;; mi; end;
F := proc(n) option remember if n=1 then return(3); fi; if n=2 then xa(3); else xa(F(n-1)); fi; end;

Mathematica

P[p_, x_] := x^2 - p x + p^2;
A280717[1] = 3;
A280717[n_] := A280717[n] = P[A280717[n - 1], NestWhile[# - 1 &, A280717[n - 1] - 1, # > A280717[n - 1]/2 && ! PrimeQ@P[A280717[n - 1], #] &]];
A280717 /@ Range[5] (* Davin Park, Feb 06 2017 *)

Python

from _future_ import division
from sympy import isprime
A280717_list, n = [3], 3
for _ in range(10):
    for i in range(1,n//2+1):
        j = i**2+n*(n-i)
        if isprime(j):
            n = j
            A280717_list.append(n)
            break # Chai Wah Wu, Jan 09 2017

Extensions

a(5) corrected and a(6)-a(9) added by Chai Wah Wu, Jan 09 2017