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Supercongruences and binary quadratic forms
Acta Arithmetica, 2021Summary: Let \(p > 3\) be a prime, and let \(a,b\) be two rational \(p\)-adic integers. We present general congruences for \(\sum_{k=0}^{p-1}\binom{a}{k}\binom{-1-a}{k}\frac{p}{k+b}\pmod{p^2} \). Let \(\{D_n\}\) be the Domb numbers given by \(D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}\binom{2n-2k}{n-k} \).
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Diagonal quadratic forms representing all binary diagonal quadratic forms
The Ramanujan Journal, 2017For a positive integer \(n\), a positive definite integral quadratic form is \(n\)-universal if it represents all positive definite integral quadratic forms of rank \(n\). For example, the quinary quadratic form \(x_1^2+x_2^2+x_3^2+x_4^2+ x_5^2\) is \(2\)-universal, by a classical result of \textit{L. J. Mordell} [Q. J. Math., Oxf. Ser.
Ji, Yun-Seong +2 more
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Indecomposable binary quadratic forms
Archiv der Mathematik, 1991The paper contains the following proposition: Let \(d>4\) and \(d\equiv 2 \bmod 4\) then there does not exist an indecomposable definite binary quadratic form with discriminant \(d\) iff the number of classes in each genus of binary quadratic forms with discriminant \(d\) is 1.
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GONI: PRIMES REPRESENTED BY BINARY QUADRATIC FORMS
2014We list 2779 regular primitive positive definite integral binary quadratic forms, and show that, conditional on the Generalized Riemann Hypothesis, this is the complete list of regular, positive definite binary integral quadratic forms (up to SL2(Z)-equivalence).
Clark, Pete L. +3 more
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Exceptional Integers for Binary Quadratic Forms
Journal of Mathematical Sciences, 2001Let \(C\) be a class of primitive binary quadratic forms, and let \(R\) be the genus containing \(C\). A number is said to be exceptional for \(C\) if it is representable by a form in \(R\) but not by a form in \(C\). The author establishes an asymptotic distribution law for exceptional numbers which extends an earlier result of \textit{O. M. Fomenko} [
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1986
An integral function is a sum of the form $$ F = \Sigma {c_{{\alpha _1} \cdots {\alpha _n}}}x_1^{{\alpha _1}} \ldots x_n^{{\alpha _n}} $$ where $$ {c_{{\alpha _1}}}{ \ldots _{{\alpha _1}}} $$ is an integer constant and \({\alpha _1}, \ldots ,{\alpha _n}\) are non negative integers.
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An integral function is a sum of the form $$ F = \Sigma {c_{{\alpha _1} \cdots {\alpha _n}}}x_1^{{\alpha _1}} \ldots x_n^{{\alpha _n}} $$ where $$ {c_{{\alpha _1}}}{ \ldots _{{\alpha _1}}} $$ is an integer constant and \({\alpha _1}, \ldots ,{\alpha _n}\) are non negative integers.
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Composition of Binary Quadratic Forms
Resonance, 2019In 2004, Bhargava introduced a new way to understand the composition law of integral binary quadratic forms through what he calls the ‘cubes of integers’. The goal of this article is to introduce the reader to Bhargava’s cubes and this new composition law, as well as to relate it to the composition law as it is known classically.
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The Representation of Binary Quadratic Forms by Positive Definite Quaternary Quadratic Forms
Transactions of the American Mathematical Society, 1994According to Dickson we call a positive definite integral quadratic form \(f\) on \(\mathbb{Z}^ n\) regular, if for every positive integer \(a\) a local representation of \(a\) by \(f\) at all completions of \(\mathbb{Z}^ n\) implies global representation of \(a\) by \(f\).
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