A145016 Primes p of the form 4k+1 for which p - floor(sqrt(p))^2 is a square.

5, 13, 17, 29, 37, 53, 73, 97, 101, 109, 137, 173, 197, 229, 241, 257, 281, 293, 349, 397, 401, 409, 457, 509, 577, 601, 641, 661, 677, 701, 733, 809, 857, 877, 977, 997, 1033, 1049, 1093, 1153, 1181, 1229, 1289, 1297, 1321, 1373, 1433, 1453, 1493, 1601, 1609

Offset: 1

Author

Vladimir Shevelev, Sep 29 2008

Keywords

• nonn
• easy

Comments

If a(n) = x^2 + y^2 then y = floor(sqrt(a(n))) and by a well known Euler theorem, the representation is unique.

Odd primes p = x^2 + y^2 such that y > x^2/2. - Thomas Ordowski, Aug 16 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A002144 (Pythagorean primes).

Programs

Maple

filter:= p -> isprime(p) and issqr(p - floor(sqrt(p))^2):
select(filter, [seq(p,p=1..10000,4)]); # Robert Israel, Dec 04 2018

Mathematica

okQ[n_]:=PrimeQ[n]&&IntegerQ[Sqrt[n-Floor[Sqrt[n]]^2]]; Select[4Range[500]+1,okQ]  (* Harvey P. Dale, Mar 23 2011 *)

PARI

isok(p) = isprime(p) && ((p%4) == 1) && issquare(p - sqrtint(p)^2); \\ Michel Marcus, Dec 04 2018