A309499 Primes p having a record value of least k such that 4*k^2*p^2 + 1 is prime.

2, 11, 17, 19, 283, 919, 1093, 1787, 9521, 181243, 257611, 274243, 857419, 1644871, 3111607, 6027277, 10452083, 14490703, 36102991, 47352131, 121431767, 171236887, 339934099, 584698243, 1177972427, 3008777311, 3091999399

Offset: 1

Author

Amiram Eldar, Aug 05 2019

Keywords

• nonn
• more

Comments

The corresponding record values of k are 1, 3, 5, 15, 20, 22, 24, 45, 95, 104, 115, 116, 135, 143, 155, 165, 179, 186, 190, 245, 250, 260, 277, 284, 310, 313, 335, ...

Gagola calculated the values of k for all the primes below 5000, and noticed that the largest value of k was only 45.

Examples

			For the primes p = 2, 3, 5, and 7, 4*p^2 + 1 = 17, 37, 101, and 197 are all primes with k = 1. 11 is the first prime with a value of k = 3, since 4*1*11^2 + 1 = 45 and 4*2^2*11^2 + 1 = 1937 are both composites, and 4*3^2*11^2 + 1 = 4357 is prime.
		

Links

• Gloria Gagola, Progress on primes, News and Letters, Mathematics Magazine, Vol. 54, No. 1 (1981), p. 43.

Crossrefs

Cf. A309498.

Programs

Mathematica

a[p_] := Module[{k = 1}, While[!PrimeQ[4 * k^2 * p^2 + 1], k++]; k]; s={}; am = 0; p = 1; Do[p = NextPrime[p]; a1 = a[p]; If[a1 > am, am=a1; AppendTo[s, p]], {n, 1, 20000}]; s