A167276 Primes p such that p^2=x^2+y^2-1 with x and y also prime.

7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 67, 73, 83, 89, 103, 107, 109, 137, 149, 151, 157, 163, 173, 191, 193, 227, 229, 233, 241, 263, 269, 293, 307, 311, 313, 317, 331, 337, 353, 359, 383, 389, 397, 401, 421, 431, 439, 443, 457, 463, 467, 487, 499, 523, 557, 577, 593, 599, 613, 619, 643, 683, 701, 727, 733, 757, 773, 829, 839, 853, 857, 863, 887, 947, 967, 977, 983, 997

Offset: 1

Author

Juri-Stepan Gerasimov, Nov 01 2009

Keywords

• nonn

Comments

Appears to be infinite.

Since (5*x+13)^2 + 1 = (3*x+7)^2 + (4*x+11)^2, it appears that there are infinitely many members of this sequence of the form 5*x+13 where x is an even number, that is the form of A030431(n). See the solution 78 at page 49 in the given reference (250 Problems in Elementary Number Theory) for the related conjecture. - Altug Alkan, Mar 30 2016

Examples

			a(1)=7 (x=5, y=5); a(2)=13 (x=7, y=11); a(3)=17 (x=11, y=17); a(4)=23 (x=13, y=19); a(5)=31 (x=11, y=31);...; a(21)=463 (x=461, y=43)
		

References

• W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, Problem 78 page 7.

Crossrefs

Cf. A000040.

Programs

Maple

isA045636 := proc(n) local p,q ; p := 2 ; while p^2+4 <= n do q := p ; while p^2+q^2 <= n do if q^2+p^2 = n then return true; end if ; q := nextprime(q) ; end do ; p := nextprime(p) ; end do ; return false ; end proc: A066872 := proc(n) ithprime(n)^2+1 ; end: for n from 1 to 200 do if isA045636(A066872(n)) then printf("%d,",ithprime(n)) ; end if ; end do ; # R. J. Mathar, Nov 09 2009

Mathematica

Select[Prime@ Range@ 168, Resolve[Exists[{x, y}, Reduce[#^2 == x^2 + y^2 - 1, {x, y}, Primes]]] &] (* Michael De Vlieger, Mar 30 2016 *)

Formula

{ A000040(i): A066872(i) in A045636}. [R. J. Mathar, Nov 09 2009]

Extensions

Edited and extended by Daniel Platt, Nov 02 2009