OFFSET
1,1
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
REFERENCES
W. SierpiĆski, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, Problem 78 page 7.
FORMULA
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Nov 01 2009
EXTENSIONS
Edited and extended by Daniel Platt, Nov 02 2009
STATUS
approved