# $p\equiv 1\pmod 4\Rightarrow p=a^2+b^2$ and $p\equiv 1\pmod 8\Rightarrow p=a^2+2b^2$, what about for $p\equiv 1\pmod {2^n}$ in general

Primes $$p$$ with $$p\equiv 1\pmod 4$$ can be written as $$p=a^2+b^2$$ for some integers $$a,b$$. For $$p\equiv 1\pmod 8$$ we have $$p=a^2+2b^2$$. Can primes that satisfy $$p\equiv 1\pmod{2^n}$$ for $$n>3$$ be written in a similar form -- for example $$p=a^2+4b^2$$ for $$n=4$$?

• for $n=4$ and $p=a^2+4b^2$: A094407 and A131204. gp-code forprime(p=3,10000,n=4;if(Mod(p,2^n)==1,if(#thue('x^2+4,p),print1(p", ")))). Aug 10, 2020 at 4:18
• For $n=4$ it's true because we can express $p$ as sum of squares and one of them should be even. Aug 10, 2020 at 8:34
• The same works for $n=5$ (consider $a^2+2b^2$ modulo 8) Aug 10, 2020 at 8:36

it gets harder, and we cannot just impose congruence conditions.

Added: one thing I've not seen in print is this: as soon as $$p \equiv 1 \pmod 8,$$ we find that $$-1$$ is a fourth power mod $$p,$$ and $$z^4 + 1 \equiv 0 \pmod p$$ has four distinct roots.

A prime can be expressed as $$p = x^2 + 32 y^2$$ if and only if $$p \equiv 1 \pmod 8$$ and $$z^4 - 2 z^2 + 2 \equiv 0 \pmod p$$ has four distinct roots.

A prime can be expressed as $$p = x^2 + 64 y^2$$ if and only if $$p \equiv 1 \pmod 8$$ and $$z^4 - 2 \equiv 0 \pmod p$$ has four distinct roots.

The 64 result can be found under "biquadratic reciprocity." Both results may be due to Gauss, but were not published until Jacobi and Eisenstein.

Let prime $$p=a^2+2^{n-2}\cdot b^2\equiv1\pmod{2^n}$$, where $$n>4$$ and $$a,b$$ is integers.

Then sequence $$(n, min(p))$$=(5,97), (6,193), (7,257), (8,257), (9,18433), (10,18433), (11,18433), (12,65537), (13,1097729), (14,65537), (15,1179649), (16,65537), (17,1179649), (18,26214401), (19,117964801), (20,26214401), (21,169869313), (22,104857601), ....

gp-code:

nminp()=
{
for(n=5,100,
forprime(p=3,10^9,
m= 2^n;
if(Mod(p,m)==1,
if(#thue('x^2+m/4, p),
print1("("n","p"), ");
break()
)
)
)
)
};