A396060 Numbers m such that m^2 + 1 and m^2 - m + 1 are both prime numbers.

2, 4, 6, 16, 90, 120, 156, 204, 210, 280, 384, 396, 406, 436, 490, 496, 636, 826, 960, 1066, 1150, 1210, 1276, 1324, 1350, 1366, 1374, 1416, 1660, 1686, 1974, 2034, 2050, 2074, 2080, 2086, 2106, 2260, 2304, 2310, 2314, 2430, 2464, 2496, 2536, 2664, 2730, 2736

Offset

1,1

Comments

Conjecture: There are infinitely many numbers m such that the two polynomials m^2 + 1 and m^2 - m + 1 are prime.

Links

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

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, Vol. 16, No. 79 (1962), pp. 363-367.

Example

6 is in the sequence because 6^2+1 = 37 and 37-6 = 31 is prime.
10 is not in the sequence because 10^2+1 = 101 is prime but 101-10 = 91 = 7*13 is not prime.

Maple

nn:=2800:
for m from 2 by 2 to nn do :
p:=m^2+1:if isprime(p) and isprime(p-m) then printf("%d, ", m):
else fi:
od:

Mathematica

Select[Range[3000], And @@ PrimeQ[#^2 + {1-#, 1}] &] (* Amiram Eldar, May 15 2026 *)

Prog

(PARI) isok(m) = isprime(m^2+1) && isprime(m^2-m+1); \\ Michel Marcus, May 16 2026

Crossrefs

Intersection of A005574 and A055494.

Sequence in context: A261864 A071243 A112086 * A070325 A199578 A367127

Adjacent sequences: A396057 A396058 A396059 * A396061 A396062 A396063

Keyword

nonn

Author

Michel Lagneau, May 15 2026

Status

approved