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Taken from problem 46 on Project Euler:

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

$9 = 7 + 2 \times 1^2$

$15 = 7 + 2 \times 2^2$

$21 = 3 + 2 \times 3^2$

...

It turns out that the conjecture was false.

Now, suppose the phrase "twice a square" was replaced with "two squares." In other words, this revision of the conjecture would state that all odd composites $C$ can be written in the following form:

$C=p+a^2+b^2$

Where $p$ is a prime and $a$ and $b$ can take on any non-negative integer values. Making the substitution $a=b$ reduces this problem to the classical conjecture.

Now, is this weaker conjecture true?


My Attempt at the Problem:

Let $d=a^2+b^2$. A well-known theorem states that the sum of two squares can be rewritten as $2^{e_0} \left( {p_1}^{e_1}...{p_n}^{e_n} \right) \left( {q_1}^{2 o_1}...{q_n}^{2 o_n} \right)$, where $p_i$ is a prime of the form $4k+1$ and $q_i$ is a prime of the form $4k+3$. We can therefore write the conjecture as follows:

$C=p+d$

$C=p+ 2^{e_0} \left( {p_1}^{e_1}...{p_n}^{e_n} \right) \left( {q_1}^{2 o_1}...{q_n}^{2 o_n} \right)$

$C=p+ 2^{e_0}g$

Where $g$ is any integer such that $g=1 \left( \text{mod } 4 \right)$. Letting $p=2$ and $e_0=0$ shows us that the conjecture is true when $C$ is of the form $4k+3$.

From here I am not sure where to go. Any clues?

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1 Answer 1

Yes, this conjecture is true. Moreover, Hua in $1938$ proved that if $n$ is odd and $n\ne 2 (\mod 3),$ then $n=p_1^2+p_2^2+p$ where $p_1$ and $p_2$ are primes. Unfortunately, all methods to prove such results are far from being elementary.

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Do you have a link or reference to this paper? And how exactly is this conjecture proven for all composite numbers - is it through some sort of extension of Hua's method? –  Ryan Sep 1 '13 at 2:54
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The result stated here does not imply the conjectured result in the question, which refers to all odd numbers. –  John Bentin Sep 1 '13 at 5:41
    
I'm wondering if leshik meant to say "This conjecture is not true"? Not quite sure. –  Ryan Sep 1 '13 at 21:17
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@Ryan The paper is "Some results in the additive prime number theory." I can't find a free link to the full text, but from a preview it appears the the result was proven for almost all $n$ satisfying the above. Regardless, this appears to neither prove nor disprove your conjecture, since it only considers the cases $n \equiv 1,3 \bmod 6$. –  Jaycob Coleman Sep 3 '13 at 3:25

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