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%I #15 Mar 30 2016 09:48:24
%S 7,13,17,23,31,37,41,43,47,53,67,73,83,89,103,107,109,137,149,151,157,
%T 163,173,191,193,227,229,233,241,263,269,293,307,311,313,317,331,337,
%U 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
%N Primes p such that p^2=x^2+y^2-1 with x and y also prime.
%C Appears to be infinite.
%C 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
%D W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, Problem 78 page 7.
%F { A000040(i): A066872(i) in A045636}. [_R. J. Mathar_, Nov 09 2009]
%e 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)
%p 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
%t Select[Prime@ Range@ 168, Resolve[Exists[{x, y}, Reduce[#^2 == x^2 + y^2 - 1, {x, y}, Primes]]] &] (* _Michael De Vlieger_, Mar 30 2016 *)
%Y Cf. A000040.
%K nonn
%O 1,1
%A _Juri-Stepan Gerasimov_, Nov 01 2009
%E Edited and extended by Daniel Platt, Nov 02 2009
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