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December 10

More on the above conjecture

Above I posed:

Conjecture. Every prime number can be written in one of the three forms ${\displaystyle a^{2}+b^{2},}$ ${\displaystyle 2a^{2}+b^{2}}$ and ${\displaystyle |2a^{2}-b^{2}|.}$

If true, it implies no natural prime is a prime in the ring ${\displaystyle \mathbb {Z} }$.

The absolute-value bars are not necessary. A number that can be written in the form ${\displaystyle -(2a^{2}-b^{2})}$ is also expressible in the form ${\displaystyle +(2a^{2}-b^{2}).}$

It turns out (experimentally; no proof) that a number that can be written in two of these forms can also be written in the third form. The conjecture is not strongly related to the concept of primality, as can be seen in this reformulation:

Conjecture. A natural number that cannot be written in any one of the three forms ${\displaystyle a^{2}+b^{2},}$ ${\displaystyle 2a^{2}+b^{2}}$ and ${\displaystyle 2a^{2}-b^{2}}$ is composite.

The first few numbers that cannot be written in any one of these three forms are

${\displaystyle 15,}$ ${\displaystyle 21,}$ ${\displaystyle 30,}$ ${\displaystyle 35,}$ ${\displaystyle 39,}$ ${\displaystyle 42,}$ ${\displaystyle 55,}$ ${\displaystyle 60,}$ ${\displaystyle 69,}$ ${\displaystyle 70,}$ ${\displaystyle 77,}$ ${\displaystyle 78,}$ ${\displaystyle 84,}$ ${\displaystyle 87,}$ ${\displaystyle 91,}$ ${\displaystyle 93,}$ ${\displaystyle 95.}$

They are indeed all composite, but why this should be so is a mystery to me. What do ${\displaystyle 2310=2\times 3\times 5\times 7\times 11,}$ ${\displaystyle 5893=71\times 83}$ and ${\displaystyle 7429=17\times 19\times 23,}$ which appear later in the list, have in common? I see no pattern.

It seems furthermore that the primorials, starting with ${\displaystyle 5\#=30,}$ make the list. (Checked up to ${\displaystyle 37\!\#=7420738134810.}$)  --Lambiam 19:23, 10 December 2024 (UTC)

Quick note, for those like me who are curious how numbers of the form ${\displaystyle -(2a^{2}-b^{2})}$ can be written into a form of ${\displaystyle 2a^{2}-b^{2}}$, note that ${\displaystyle 2a^{2}-b^{2}=(2a+b)^{2}-2(a+b)^{2}}$, and so ${\displaystyle 2a^{2}-b^{2}=-p\Rightarrow p=2(a+b)^{2}-(2a+b)^{2}}$. GalacticShoe (talk) 02:20, 11 December 2024 (UTC)

A prime is expressible as the sum of two squares if and only if it is congruent to ${\displaystyle 1\!\!\!{\pmod {4}}}$, as per Fermat's theorem on sums of two squares. A prime is expressible of the form ${\displaystyle 2a^{2}+b^{2}}$ if and only if it is congruent to ${\displaystyle 1,3\!\!{\pmod {8}}}$, as per OEIS:A002479. And a prime is expressible of the form ${\displaystyle 2a^{2}-b^{2}}$ if and only if it is congruent to ${\displaystyle 1,7\!\!{\pmod {8}}}$, as per OEIS:A035251. Between these congruences, all primes are covered. GalacticShoe (talk) 05:59, 11 December 2024 (UTC)

More generally, a number is not expressible as:

1. ${\displaystyle a^{2}+b^{2}}$ if it has a prime factor congruent to ${\displaystyle 3\!\!\!{\pmod {4}}}$ that is raised to an odd power (equivalently, ${\displaystyle 3,7\!\!{\pmod {8}}}$.)
2. ${\displaystyle 2a^{2}+b^{2}}$ if it has a prime factor congruent to ${\displaystyle 5,7\!\!{\pmod {8}}}$ that is raised to an odd power
3. ${\displaystyle 2a^{2}-b^{2}}$ if it has a prime factor congruent to ${\displaystyle 3,5\!\!{\pmod {8}}}$ that is raised to an odd power

It is easy to see why expressibility as any two of these forms leads to the third form holding, and also we can see why it's difficult to see a pattern in numbers that are expressible in none of these forms, in particular we get somewhat-convoluted requirements on exponents of primes in the factorization satisfying congruences modulo 8. GalacticShoe (talk) 06:17, 11 December 2024 (UTC)

Thanks. Is any of this covered in some Misplaced Pages article?  --Lambiam 10:06, 11 December 2024 (UTC)

All primes? 2 is not covered! 176.0.133.82 (talk) 08:00, 17 December 2024 (UTC)

${\displaystyle 2}$ can be written in all three forms: ${\displaystyle 2=1^{2}+1^{2}=2\cdot 1^{2}+0^{2}=2\cdot 1^{2}-0^{2}.}$  --Lambiam 09:38, 17 December 2024 (UTC)

I don't say it's not covered by the conjecture. I say it's not covered by the discussed classes of remainders. 176.0.133.82 (talk) 14:54, 17 December 2024 (UTC)

Odd prime, my bad. GalacticShoe (talk) 16:38, 17 December 2024 (UTC)

Assume p is 3 mod 4. Suppose that (2|p)=1. Then ${\displaystyle x^{4}+1\equiv (x^{2}+\lambda x+1)(x^{2}-\lambda x+1){\pmod {p}}}$ where ${\displaystyle \lambda ^{2}-2\equiv 0}$. Because the cyclotomic ideal ${\displaystyle (p,\zeta ^{2}+\lambda \zeta +1)}$ has norm ${\displaystyle p^{2}}$ and is stable under the Galois action ${\displaystyle \zeta \mapsto 1/\zeta }$ it is generated by a single element ${\displaystyle a\zeta ^{2}+b\zeta +a}$, of norm ${\displaystyle (2a^{2}-b^{2})^{2}}$.

If (2|p)=-1, then the relevant ideal is stable under ${\displaystyle \zeta \mapsto -1/\zeta }$ and so is generated by ${\displaystyle a\zeta ^{2}+b\zeta -a}$, of norm ${\displaystyle (2a^{2}+b^{2})^{2}}$. Tito Omburo (talk) 14:43, 11 December 2024 (UTC)