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      Questions tagged [quadratic-forms]

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      2
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      1answer
      120 views

      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 ...
      5
      votes
      1answer
      93 views

      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 $\alpha$, ...
      0
      votes
      0answers
      249 views

      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 ...
      4
      votes
      2answers
      199 views

      Quadratic diophantine equations and geometry of numbers

      Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system $$ w^2 - ax^2 -by^2 + abz^2 = 1 $$ $$ \...
      1
      vote
      0answers
      157 views

      Topological vs algebraic intersection forms

      Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...
      1
      vote
      0answers
      68 views

      Proper ideals are invertible

      I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma: Lemma 7.5: Let $...
      3
      votes
      1answer
      215 views

      Strong Approximation for sol'ns to quadratic diophantine equations

      Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true: For any 4-tuple $\xi =...
      7
      votes
      3answers
      253 views

      Integer positive definite quadratic form as a sum of squares

      Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$...
      2
      votes
      2answers
      191 views

      Solving diagonal simultaneous quadratic equations

      A problem I am trying to solve has led to me to the following system of equations: $$A(x^2) + Bx + c = 0$$ Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
      1
      vote
      0answers
      87 views

      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 ...
      9
      votes
      1answer
      196 views

      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 ...
      0
      votes
      0answers
      107 views

      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 ...
      1
      vote
      0answers
      89 views

      Nonnegative integers written as $x(3x+1)+y(3y-1)+z(3z+2)+w(3w-2)$ with $x,y,z,w\in\{0,1,2,\ldots\}$

      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+...
      3
      votes
      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, ...
      5
      votes
      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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