OFFSET
1,1
COMMENTS
Gauss proved that any prime of the form 4*k+1 (A002144) is equal to the unique sum of two squares. This sequence identifies these summed squares and uniquely maps them into a decimal from which the two squares can be retrieved. The mapping is given by a(n) = 2^x+2^y where p(n) = x^2+y^2.
LINKS
L. A. Butler, A Classification of Gaussian Primes, School of Mathematics, Bristol Univ.
EXAMPLE
p(6) = 41 and 41 = 4^2 + 5^2 hence a(6) = 2^4+2^5 = 48. To retrieve the values 4 and 5 from 48 convert 48 to binary. The 1 bits (there are only ever two) select 2^4 and 2^5. So x, y are 4, 5.
MATHEMATICA
next1m4prime[n1_] := (n2=n1+1; While[!PrimeQ[n2]||!Mod[n2, 4]==1, n2++]; n2); getbinmap[m1_] := (m2=m1; m3=Floor[Sqrt[m2]]; While[!IntegerQ@Sqrt[m2-m3^2], m3--]; 2^Sqrt[m2-m3^2] + 2^m3); SetAttributes[getbinmap, Listable]; getbinmap[Table[Nest[next1m4prime, 1, n], {n, 1, 100}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Aug 19 2013
STATUS
approved