# Tag Info

Accepted

### Lagrange four squares theorem

The set $X$ doesn't have to be the set of non-negative integers. This was known already to Härtter and Zöllner in 1977, who constructed an $X$ of the form $\{ 0, 1, 2, \ldots \} \setminus S$ for an ...
• 9,039
Accepted

• 71.6k
Accepted

### Many representations as a sum of three squares

The formula for $r_3(n)$ essentially connects this with a class number of an imaginary quadratic field, or (apart from the $\sqrt{n}$ scaling) with the value of an $L$-function at $1$. So your ...
• 42.4k
Accepted

### Lagrange four-squares theorem --- deterministic complexity

As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a ...
• 28.7k

### Many representations as a sum of three squares

Let me restrict to the number of primitive representations $$r_3^\ast(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n\ \text{and}\ \gcd(a,b,c)=1 \}\right|.$$ Note that $r_3(n)$ can be easily ...
• 85.6k

• 9,039

### Jacobi's theorem on sums of two squares (reference request)

Using class field theory one can prove for every integer $n>0$ there exists a monic polynomial $f$ of degree $h(-4n)$* such that for any odd integer $m$ coprime to $n$ we have the following ...
• 2,371

### Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

Answer to Question 1. Yes, $E(\mathbb{Z})=F$. The inclusion $F\subseteq E(\mathbb{Z})$ follows from $E\subseteq E(\mathbb{Z})$ and Dickson's result $E=F$. It remains to show $E(\mathbb{Z})\subseteq F$...
• 2,519

### $x^2+7y^2=2^n$ and sums of four squares

The Diophantine equation $x^2 + 7y^2 = 2^n$ is "trivial" in the sense that its solutions can be described in a very simple way (see boxed formula near the end for solutions in odd numbers ...
• 41.9k
Accepted

### Representing $x^3-2$ as a sum of two squares

The answer is similar to one provided here. The approach is elementary and proves stronger fact that it can be expressed as sum of two coprime squares. We consider the product $(n+2)(n^3-2)$ which is ...
• 878

### A generalization of partition function to the sums of squares

This asymptotic was stated by Hardy and Ramanujan, but without proof. The first proof of this asymptotic was given by Wright in 1934 [1], by a rather complicated argument. A much simpler approach ...
• 5,794

### For a proof of the three-square theorem without using Dirichlet's theorem on primes in arithmetic progressions

I am not sure if Legendre's proof was complete (see this related MO entry), but surely Gauss gave a proof without Dirichlet's theorem (which came decades later). A nice transparent proof (without ...
• 85.6k
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• 6,209