1

We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$.

Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be represented written in an unique way as $ga^2+hb^2+ic^2$ where $g,h,i\in\Bbb Z$, $a,b,c,d,r\in\Bbb N$"?

At least what are some of famous ones?

I think it will be good to have such a detailed list and some references.

  • See http://arxiv.org/pdf/1207.0172.pdf. As the paper says (ending of page 6), all the representations in Theorem 1.1 are unique. – user236182 Sep 05 '15 at 23:02
  • Uniqueness of $1.2,1.3$ known? –  Sep 05 '15 at 23:11
  • Known for ($1.2$) $(I)$ (see here). $(II)$ is not unique, e.g. $23=2\cdot 4^2-3^2=2\cdot 6^2-7^2$, same for $(III)$: $11=3\cdot 3^2-4^2=3\cdot 5^2-8^2$. Known for ($1.3$) $(I)$ (same as $1.2$ $(I)$), not unique for $(II)$: $17=5^2-2\cdot 2^2=7^2-2\cdot 4^2$ and $(III)$: $37=7^2-3\cdot 2^2=8^2-3\cdot 3^2$. – user236182 Sep 05 '15 at 23:21
  • $1.2$: $(I), (IV), (IX)$ are unique, all others are not unique. Same for $1.3$. – user236182 Sep 05 '15 at 23:35

1 Answers1

0

See http://arxiv.org/pdf/1207.0172.pdf.

As ending of page $6$ says, all representations in Theorem $1.1$ are unique.

$1.2$: $(I), (IV), (IX)$ are unique, all the others aren't. Same for $1.3$.

To prove uniqueness, simply observe $$k^2a^2-b^2=(ka+b)(ka-b)=p\iff \begin{cases}ka+b=p\\ ka-b=1\end{cases}$$

for all $k\in\{1,2,3\}, a,b\ge 0$. Same for $a^2-k^2b^2$.

To disprove uniqueness, find counterexamples with a program.

$1.4$: all of them are unique for $p\le 10\, 000$.

user236182
  • 13,324