A090688 First occurrence of primes in the progression k*x^2-1.

3, 7, 2, 3, 19, 5, 251, 7, 89, 43, 11, 467, 13, 59, 67, 17, 683, 19, 83, 197, 367, 23, 103, 107, 251, 463, 29, 4463, 31, 131, 1223, 139, 11987, 37, 7643, 359, 163, 41, 13931, 43, 179, 33533, 751, 47, 199, 5099, 467, 211, 53, 1979, 223, 227, 521, 23599, 59, 8783, 61

Offset: 1

Author

Cino Hilliard, Dec 18 2003

Keywords

• easy
• nonn

Comments

It appears that the generating function k*x^2-1 will produce all primes eventually with some repetitions.

If k>2 is a square, there is no entry corresponding to k*x^2-1. Bunyakovsky's conjecture implies that there are primes for all other k. - Robert Israel, Nov 22 2020

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

f:= proc(k) local x;
  if issqr(k) then return NULL fi;
  for x from 1 do if isprime(k*x^2-1) then return k*x^2-1 fi od
end proc:
f(1):= 3: f(4):= 3:
map(f, [$1..300]); # Robert Israel, Nov 22 2020

Formula

If p>=5 is prime, a(p+3-floor(sqrt(p)))=p. - Robert Israel, Nov 22 2020