O rozkladu čísel v součet deseti a dvanácti čtverců
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Karel Petr (1905)
Časopis pro pěstování mathematiky a fysiky
J. Brzezinski (1991)
Commentarii mathematici Helvetici
Lomadze, G. (2004)
Georgian Mathematical Journal
Elise Björkholdt (2000)
Journal de théorie des nombres de Bordeaux
Let be a totally real algebraic number field whose ring of integers is a principal ideal domain. Let be a totally definite ternary quadratic form with coefficients in . We shall study representations of totally positive elements by . We prove a quantitative formula relating the number of representations of by different classes in the genus of to the class number of , where is a constant depending only on . We give an algebraic proof of a classical result of H. Maass on representations...
Toyokazu Hiramatsu, Yoshio Mimura (1990)
Acta Arithmetica
Lomadze, G. (1999)
Georgian Mathematical Journal
Lomadze, G. (1994)
Georgian Mathematical Journal
Fan Ge, Zhi-Wei Sun (2016)
Colloquium Mathematicae
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in...
Zhi-Wei Sun (2012)
Acta Arithmetica
Zhi-Wei Sun (2012)
Colloquium Mathematicae
Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum , where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and , then , where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If then .
James W. Cogdell (2003)
Journal de théorie des nombres de Bordeaux
We address the question of when an integer in a totally real number field can be written as the sum of three squared integers from the field and more generally whether it can be represented by a positive definite integral ternary quadratic form over the field. In recent work with Piatetski-Shapiro and Sarnak we have shown that every sufficiently large totally positive square free integer is globally integrally represented if and only if it is so locally at all places, thus essentially resolving...
Jan Wójcik (1971)
Colloquium Mathematicae
Jerzy Browkin (2010)
Colloquium Mathematicae
We consider systems of equations of the form and , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.
Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesús M. Ruiz (2010)
Bulletin de la Société Mathématique de France
We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...
Pierre Kaplan, Kenneth S. Williams (2004)
Acta Arithmetica
Ernest X. W. Xia (2014)
Colloquium Mathematicae
For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form . Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.
Lomadze, G. (1998)
Georgian Mathematical Journal
Lomadze, G. (1995)
Georgian Mathematical Journal
Zhi-Hong Sun, Kenneth S. Williams (2006)
Acta Arithmetica
Zhi-Hong Sun (2011)
Acta Arithmetica
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