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# Questions tagged [quadratic-forms]

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### Representation of two related integers by the same binary quadratic form

Let $f(x,y) = ax^2 + bxy - cy^2$ be an indefinite, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square ...
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### Evaluating a binary quadratic form at convergents

We use the notation $$\displaystyle [a_0; a_1, \cdots, a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}$$ to denote a finite continued fraction, and for a given real number $\alpha$, ...
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### On positive integers of the forms $x^2+5y^2$, $x^2+6y^2$, $x^2+10y^2$

For a positive integer $n$ and a prime $p$, we let $\nu_p(n)$ denote the $p$-adic valuation of $n$ (i.e., the $p$-adic order of $n$). On the basis of my computation, I formulated the following ...
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### Norm quadrics and their motives

Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...
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### Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary. Baeza [B, V.6.3] extended this result to Witt ...
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### A conjecture stronger than Gauss' triangular number theorem

A subset $A$ of $\mathbb N=\{0,1,2,\ldots\}$ is called an asymptotic additive basis of order $3$ if any sufficiently large integer can be written as the sum of three elements of $A$. I have ...
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Recently, I formulated the following conjecture which is somewhat similar to Lagrange's four-square theorem. Conjecture. Each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x(3x+1)+y(3y-1)+z(3z+... 0answers 146 views ### On sums of three squares Let$\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$. Clearly, ... 0answers 207 views ### Sums of two integer squares in arithmetic progressions Is there an explicit formula in the literature for the number of representations of a positive integer$n$as a sum of two integer squares, the second of which is divisible by$5\$? So this means to ...

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