All Questions
Tagged with sums-of-squares quadratic-forms
11 questions
4
votes
1
answer
148
views
Witt ring of a field with Pythagoras number $2$
I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...
- 272
-1
votes
1
answer
252
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Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$
Question 1. My computation in 2018 suggests that a sum of two integer squares is a sum of three nonzero integer squares if and only if it is not of the form
$$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,...
- 15.6k
-1
votes
1
answer
182
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$7n=x^2+2y^2+4z^2$ with or without $x^2\equiv y^2\equiv z^2\pmod 7$
It is well known that any positive odd integer can be written as $x^2+2y^2+4z^2$ with $x,y,z\in\mathbb Z$.
Question 1. Whether for any odd integer $n>93$ there are $x,y,z\in\mathbb Z$ such that $7n=...
- 15.6k
3
votes
0
answers
203
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On sums of three squares
Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, ...
- 15.6k
5
votes
0
answers
229
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Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
- 15.6k
16
votes
0
answers
588
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The number 1680 and Lagrange's four-square theorem
The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...
- 15.6k
12
votes
2
answers
499
views
"Pythagoras number" for integral matrices
It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $...
- 3,031
18
votes
2
answers
3k
views
Many representations as a sum of three squares
Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...
- 1,072
3
votes
1
answer
159
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Sum of Squares Length of a Product
Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...
- 3,031
1
vote
2
answers
252
views
Impossible Range for Minkowski-Like Sum of Squares
Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of $Mx^2-N(a^2+b^2+c^2)$, is there any such z which is "deceptive", meaning that it ...
- 1,547
7
votes
3
answers
1k
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Realizing proper pure octonions as conjugates
Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ
We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...
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