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

Author

Chris Fry, Nov 29 2013

Keywords

• nonn
• easy

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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 *)