A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime.

2, 12, 14, 24, 34, 122, 154, 164, 272, 342, 464, 612, 674, 734, 784, 794, 854, 1174, 1262, 1274, 1364, 1392, 1524, 1554, 1664, 1682, 1844, 1854, 1862, 1892, 1924, 1942, 1994, 2232, 2294, 2354, 2442, 2592, 2802, 2884, 3124, 3164, 3292, 3394, 3544, 3594, 3632, 3724, 3892, 3904, 3922

Offset

1,1

Comments

See A027862 for primes of the form x^2+(x+1)^2 = 2x^2+2x+1.

See A027864 for primes of the form x^2+(x+1)^2+(x+2)^2 = 3x^2+6x+5.

It is an open question whether either of these polynomials produces an infinite number of primes. This sequence lists the values of x that produce a prime in both polynomials. x must be congruent to 0 or 2 (mod 4) and all the generated primes are of the form 4k+1.

References

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 266.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Wikipedia, Hardy and Littlewood's Conjecture F.

Example

When x=14, 2x^2+2x+1=421 and 3x^2+6x+5=677. 14 is the third value of x for which both these polynomials produce a prime number, so a(3)=14.

Mathematica

lst = {}; Do[If[And[PrimeQ[n^2 + (n + 1)^2], PrimeQ[n^2 + (n + 1)^2 + (n + 2)^2]], Print[n]; AppendTo[lst, n]], {n, 10000}]
Select[Range[2, 4000, 2], AllTrue[{(#^2+(#+1)^2), (#^2+(#+1)^2+(#+2)^2)}, PrimeQ]&] (* Harvey P. Dale, Jul 30 2023 *)

Crossrefs

Cf. A027862, A027864. Equals n common to A027861 and A027863.

Sequence in context: A124163 A108976 A073598 * A022368 A076484 A259128

Adjacent sequences: A232765 A232766 A232767 * A232769 A232770 A232771

Keyword

nonn,easy

Author

Chris Fry, Nov 29 2013

Status

approved