A sum of three nonzero triangular numbers

2021 ◽  
pp. 1-22
Author(s):  
Daejun Kim ◽  
Jeongwon Lee ◽  
Byeong-Kweon Oh
Keyword(s):  
Riemann Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Triangular Numbers ◽  
Finite Set ◽  
Polygonal Numbers ◽  
Sums Of Three Squares ◽  
Number Of Authors

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as [Formula: see text] for some integers [Formula: see text]. Furthermore, we explicitly describe this finite set under the GRH. From this result, we also describe further generalizations for sums of nonzero polygonal numbers. Precisely, we find all integers, under the GRH only when [Formula: see text], which are sums of [Formula: see text] nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer [Formula: see text].

scholarly journals On the bounded generation of arithmetic SL2

2019 ◽  
Vol 116 (38) ◽  
pp. 18880-18882 ◽  
Author(s):  
Bruce W. Jordan ◽  
Yevgeny Zaytman
Keyword(s):  
Number Theory ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Additional Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Group Of Units ◽  
Finite Set ◽  
Bounded Generation

Let K be a number field and S be a finite set of primes of K containing the archimedean valuations. Let 𝒪 be the ring of S-integers in K. Morgan, Rapinchuck, and Sury [A. V. Morgan et al., Algebra Number Theory 12, 1949–1974 (2018)] have proved that if the group of units 𝒪× is infinite, then every matrix in SL2(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that K has at least 1 real embedding or S contains a finite place we can get a product of at most 8 elementary matrices. If we assume a suitable generalized Riemann hypothesis, then every matrix in SL2(𝒪) is the product of at most 5 elementary matrices if K has at least 1 real embedding, the product of at most 6 elementary matrices if S contains a finite place, and the product of at most 7 elementary matrices in general.

scholarly journals I. Continuation of the subject of a paper read Dec. 22, 1853, the supplement to which was read Jan. 12, 1854, by Sir Frederick Pollock, &c.; with a proof of Fermat's first and second theorems of the polygonal numbers, viz. that every odd number is composed of four square numbers or less, and of three triangular numbers or less

1856 ◽  
Vol 7 ◽  
pp. 1-4
Keyword(s):  
Triangular Numbers ◽  
The Subject ◽  
Polygonal Numbers ◽  
Four Square

The object of this paper is in the first instance to prove the truth of a theorem stated in the supplement to a former paper, viz. “that every odd number can be divided into four squares (zero being considered an even square) the algebraic sum of whose roots (in some form or other) will equal 1, 3, 5, 7, &c. up to the greatest possible sum of the roots.” The paper also contains a proof, that if every odd number 2 n + 1 can be divided into four square numbers, the algebraic sum of whose roots is equal to 1, then any number n is composed of not exceeding three triangular numbers.

Quadratic Non-Residues and Prime-Producing Polynomials

1989 ◽  
Vol 32 (4) ◽  
pp. 474-478 ◽  
Author(s):  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Riemann Hypothesis ◽  
Quadratic Polynomials ◽  
Generalized Riemann Hypothesis ◽  
Initial Values ◽  
Positive Discriminant

AbstractWe will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

scholarly journals Variant of the truncated Perron formula and primes in polynomial sets

2019 ◽  
Vol 16 (02) ◽  
pp. 309-323
Author(s):  
D. S. Ramana ◽  
O. Ramaré
Keyword(s):  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Partial Sums ◽  
Generalized Riemann Hypothesis ◽  
Perron’S Formula ◽  
Polynomial Formula

We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial [Formula: see text], for every [Formula: see text], and almost every prime [Formula: see text] in [Formula: see text], the number of primes from the interval [Formula: see text] that are values of [Formula: see text] modulo [Formula: see text] is the expected one, provided [Formula: see text] is not more than [Formula: see text]. We obtain this via a variant of the classical truncated Perron’s formula for the partial sums of the coefficients of a Dirichlet series.

scholarly journals The André–Oort conjecture for Drinfeld modular varieties

2013 ◽  
Vol 149 (4) ◽  
pp. 507-567 ◽  
Author(s):  
Patrik Hubschmid
Keyword(s):  
Riemann Hypothesis ◽  
Classical Case ◽  
Base Field ◽  
Generalized Riemann Hypothesis ◽  
Fixed Set ◽  
Special Points ◽  
Modular Varieties

AbstractWe consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

The distribution of quadratic residues and non-residues in the Goldwasser–Micali type of cryptosystem

2014 ◽  
Vol 8 (2) ◽  
Author(s):  
Benjamin Justus
Keyword(s):  
Riemann Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Quadratic Residues

Abstract.We provide unconditional results and conditional ones under the assumption of GRH (Generalized Riemann Hypothesis) on the distribution of quadratic residues and quadratic non-residues in

scholarly journals Periodic points and invariant pseudomeasures for toral endomorphisms

1986 ◽  
Vol 6 (3) ◽  
pp. 449-473 ◽  
Author(s):  
William A. Veech
Keyword(s):  
Periodic Orbit ◽  
Riemann Hypothesis ◽  
Periodic Points ◽  
One Dimension ◽  
Generalized Riemann Hypothesis

AbstractExtending a result of Livsic [10] it is proved that the coboundary equation f(Tx)−f(x) = g(x) admits a C∞ solution f for C∞g when T is an ergodic toral endomorphism and g sums to zero over every periodic orbit. The same statement is false with C1 in place of C∞, in contrast to the Livsic (hyperbolic) theorem. In one dimension the ‘Lip α’ case leads to questions relating to the generalized Riemann hypothesis.

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