scholarly journals On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor

2020 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
Bagram Sibgatullovich Kochkarev
Keyword(s):  
Diophantine Equation ◽  
Natural Numbers

The diophantine equation x2 + paqb = yq

2021 ◽  
pp. 1-18
Author(s):  
Hemar Godinho ◽  
Victor G. L. Neumann
Keyword(s):  
Diophantine Equation ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Lucas Sequences ◽  
Primitive Divisors

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

Quelques résultats sur les équations axp + byp = cz2

2006 ◽  
Vol 58 (1) ◽  
pp. 115-153 ◽  
Author(s):  
W. Ivorra ◽  
A. Kraus
Keyword(s):  
Prime Number ◽  
Diophantine Equation ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Modular Representations ◽  
Natural Numbers ◽  
Prime Divisors

AbstractLet p be a prime number ≥ 5 and a, b, c be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation axp + byp = cz2, in case the product of the prime divisors of abc divides 2ℓ, with ℓ an odd prime number. For instance, under some conditions on a, b, c, we provide a constant f (a, b, c) such that there are no such solutions if p > f (a, b, c). In application, we obtain information concerning the ℚ-rational points of hyperelliptic curves given by the equation y2 = xp + d with d ∈ ℤ.

scholarly journals On a class of quartic Diophantine equations

2021 ◽  
Vol 27 (1) ◽  
pp. 1-6
Author(s):  
F. Izadi ◽  
◽  
M. Baghalaghdam ◽  
S. Kosari ◽  
◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Nontrivial Solutions ◽  
Natural Numbers ◽  
Positive Rank

In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) { \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }, where a_i and n\geq3 are fixed arbitrary integers. We try to transform this quartic to a cubic elliptic curve of positive rank. We solve the equation for some values of a_i and n=3,4, and find infinitely many nontrivial solutions for each case in natural numbers, and show among other things, how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for n\geq 3, this paper will be restricted to the examples where n=3,4. Finally, we explain how to solve more general cases (n\geq 4) without giving concrete examples to case n\geq 5.

scholarly journals On Second Order Gap Balancing Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
S. S. Rout
Keyword(s):  
Diophantine Equation ◽  
Second Order ◽  
Natural Numbers ◽  
Gap Balancing ◽  
First Order ◽  
Balancing Numbers

We consider the Diophantine equation 1k+2k+⋯+x-1k=x+2k+x+3k+⋯+x+rk for some natural numbers x, k, and r, and we call 2x+1 as kth order 2-gap balancing number. It was also proved that there are infinitely many first order 2-gap balancing numbers. In this paper, we show that the only second order 2-gap balancing number is 1.

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

Further improvements in Waring’s problem. Ill Eighth powers

1993 ◽  
Vol 345 (1676) ◽  
pp. 385-396 ◽  
Keyword(s):  
Diophantine Equation ◽  
Exponential Sums ◽  
Natural Numbers ◽  
Prime Divisors ◽  
Waring’S Problem ◽  
Waring's Problem

In this paper we continue our development of the methods of Vaughan & Wooley, these being based on the use of exponential sums over integers having only small prime divisors. On this occasion we concentrate on improvements in the estimation of the contribution of the major arcs arising in the efficient differencing process. By considering the underlying diophantine equation, we are able to replace certain smooth Weyl sums by classical Weyl sums, and thus we are able to utilize a number of pruning processes to facilitate our analysis. These methods lead to improvements in Waring’s problem for larger k . In this instance we prove that G (8) ⩽ 42, which is to say that all sufficiently large natural numbers are the sum of at most 42 eighth powers of integers. This improves on the earlier bound G (8) ⩽ 43.

Numbers Differing from Consecutive Squares by Squares

1985 ◽  
Vol 28 (3) ◽  
pp. 337-342 ◽  
Author(s):  
E. J. Barbeau
Keyword(s):  
Natural Number ◽  
Diophantine Equation ◽  
Natural Numbers ◽  
Perfect Squares

AbstractIt is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.

scholarly journals Intersecting semi-disks and the synergy of three quadratic forms

2019 ◽  
Vol 27 (2) ◽  
pp. 5-14
Author(s):  
Andrew D. Ionaşcu
Keyword(s):  
Diophantine Equation ◽  
Quadratic Forms ◽  
Infinitely Many Solutions ◽  
Natural Numbers ◽  
Integer Variables ◽  
Geometry Problem

AbstractIn this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a2 + b2, a2 + 2b2, and 2a2 − b2. We show that there are infinitely many solutions and conjecture that there are always solutions if x ≥ 5 and x ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables.

scholarly journals Subbalancing Numbers

MATEMATIKA ◽  
2018 ◽  
Vol 34 (1) ◽  
pp. 163-171
Author(s):  
Ravi Kumar Davala ◽  
G. K. Panda
Keyword(s):  
Natural Number ◽  
Diophantine Equation ◽  
Natural Numbers ◽  
Left Hand

A natural number $n$ is called balancing number (with balancer $r$)if it satisfies the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r).$ However, if for some pair of natural numbers $(n,r)$, $1+2+\cdots+(n-1) < (n+1)+(n+2)+\cdots+(n+r)$ and equality is achieved after adding a natural number $D$ to the left hand side then we call $n$ a $D$-subbalancing number with $D$-subbalaner number $r$. In this paper, such numbers are studied for certain values of $D$.

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