login
A321871
Numbers k such that m = 4k^2 + 2k + 17 and 4m - 3 are both primes.
2
1, 3, 4, 9, 11, 14, 23, 36, 38, 49, 66, 101, 133, 134, 141, 154, 158, 191, 193, 196, 198, 213, 228, 241, 269, 283, 294, 313, 334, 339, 384, 394, 411, 413, 431, 453, 499, 511, 554, 558, 601, 619, 639, 649, 661, 686, 701, 704, 718, 758, 791, 804, 818, 821, 881
OFFSET
1,2
COMMENTS
Rotkiewicz proved that if k is in this sequence, and m = 4k^2 + 2k + 17, then m*(4m - 3) is a decagonal Fermat pseudoprime to base 2 (A321870), and thus under Schinzel's Hypothesis H there are infinitely many decagonal Fermat pseudoprimes to base 2.
The corresponding pseudoprimes are 2047, 13747, 31417, 514447, 1092547, 2746477, 18985627, 111202297, 137763037, ...
LINKS
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
EXAMPLE
1 is in the sequence since 4*1^2 + 2*1 + 17 = 23 and 4*23 - 3 = 89 are both primes.
MATHEMATICA
Select[Range[1000], PrimeQ[4#^2 + 2# + 17] && PrimeQ[16#^2 + 8# + 65] &]
PROG
(PARI) isok(n) = isprime(m=4*n^2 + 2*n + 17) && isprime(4*m-3); \\ Michel Marcus, Nov 20 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 20 2018
STATUS
approved