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
KEYWORD
nonn,easy
AUTHOR
Chris Fry, Nov 29 2013
STATUS
approved