scholarly journals The Primitive-Solutions of Diophantine Equation x^2+pqy^2=z^2, for primes p,q

2022 ◽  
Vol 18 (2) ◽  
pp. 308-314
Author(s):  
Aswad Hariri Mangalaeng
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Positive Integers

In this paper, we determine the primitive solutions of diophantine equations x^2+pqy^2=z^2, for positive integers x, y, z, and primes p,q. This work is based on the development of the previous results, namely using the solutions of the Diophantine equation x^2+y^2=z^2, and looking at characteristics of the solutions of the Diophantine equation x^2+3y^2=z^2 and x^2+9y^2=z^2.

ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

2014 ◽  
Vol 91 (1) ◽  
pp. 11-18
Author(s):  
NOBUHIRO TERAI
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Euler’S Totient Function ◽  
Exponential Diophantine Equations ◽  
Euler’S Theorem ◽  
Positive Integers

AbstractLet $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ is called the Euler quotient of $m$ with base $a$. By Euler’s theorem, $E_{m}(a)$ is an integer. In this paper, we consider the Diophantine equation $E_{m}(a)=x^{l}$ in integers $x>1,l>1$. We conjecture that this equation has exactly five solutions $(a,m,x,l)$ except for $(l,m)=(2,3),(2,6)$, and show that if the equation has solutions, then $m=p^{s}$ or $m=2p^{s}$ with $p$ an odd prime and $s\geq 1$.

scholarly journals A note on the ternary Diophantine equation x 2 − y 2 m = z n

2021 ◽  
Vol 29 (2) ◽  
pp. 93-105
Author(s):  
Attila Bérczes ◽  
Maohua Le ◽  
István Pink ◽  
Gökhan Soydan
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Special Cases ◽  
Positive Integers

Abstract Let ℕ be the set of all positive integers. In this paper, using some known results on various types of Diophantine equations, we solve a couple of special cases of the ternary equation x2 − y2m = zn , x, y, z, m, n ∈ ℕ, gcd(x, y) = 1, m ≥ 2, n ≥ 3.

A GENERALIZATION OF A THEOREM OF BUMBY ON QUARTIC DIOPHANTINE EQUATIONS

2006 ◽  
Vol 02 (02) ◽  
pp. 195-206 ◽  
Author(s):  
MICHAEL A. BENNETT ◽  
ALAIN TOGBÉ ◽  
P. G. WALSH
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers

Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X4 - 2Y2 = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integers solutions to the Diophantine equation (m2 + m + 1)X4 - (m2 + m)Y2 = 1 are (X,Y) = (1, 1),(2m + 1, 4m2 + 4m + 3).

The Diophantine equation Fn = P(x)

2020 ◽  
Vol 16 (09) ◽  
pp. 2095-2111
Author(s):  
Szabolcs Tengely ◽  
Maciej Ulas
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Infinitely Many Solutions ◽  
Polynomial Formula ◽  
Positive Integers ◽  
Integral Solutions

We consider equations of the form [Formula: see text], where [Formula: see text] is a polynomial with integral coefficients and [Formula: see text] is the [Formula: see text]th Fibonacci number that is, [Formula: see text] and [Formula: see text] for [Formula: see text] In particular, for each [Formula: see text], we prove the existence of a polynomial [Formula: see text] of degree [Formula: see text] such that the Diophantine equation [Formula: see text] has infinitely many solutions in positive integers [Formula: see text]. Moreover, we present results of our numerical search concerning the existence of even degree polynomials representing many Fibonacci numbers. We also determine all integral solutions [Formula: see text] of the Diophantine equations [Formula: see text] for [Formula: see text] and [Formula: see text].

scholarly journals A Simple Solution for Diophantine Equations of Second, Third and Fourth Power

2005 ◽  
Vol 4 (1) ◽  
pp. 96-100
Author(s):  
A. K. Maran
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Simple Solution ◽  
Fourth Power ◽  
Trial And Error ◽  
Pythagorean Triples ◽  
Trial And Error Method ◽  
Positive Integers ◽  
Error Method

We know already that the set Of positive integers, which are satisfying the Pythagoras equation Of three variables and four variables cre called Pythagorean triples and quadruples respectively. These cre Diophantine equation OF second power. The all unknowns in this Pythagorean equation have already Seen by mathematicians Euclid and Diophantine. Hcvwever the solution defined by Euclid are Diophantine is also again having unknowns. The only to solve the Diophantine equations wos and error method. Moreover, the trial and error method to obtain these values are not so practical and easy especially for time bound work, since the Diophantine equations are having more than unknown variables.

Quadratic diophantine equations

1960 ◽  
Vol 253 (1026) ◽  
pp. 227-254 ◽  
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Positive Integers

Tartakowsky (1929) proved that a positive definite quadratic form, with integral coefficients, in 5 or more variables represents all but at most finitely many of the positive integers not excluded by congruence considerations. Tartakowsky’s argument does not lead to any estimate for a positive integer which, though not so excluded, is not represented by the quadratic form. Here estimates for such an integer are obtained, in terms of the coefficients of the quadratic form. To simplify the argument and improve the estimates, the problem is slightly generalized (by considering a Diophantine equation with linear terms). A combination of analytical and arithmetical methods is needed.

scholarly journals A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

2020 ◽  
Vol 55 (2) ◽  
pp. 195-201
Author(s):  
Maohua Le ◽  
◽  
Gökhan Soydan ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Exponential Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers ◽  
Upper Bound For Solutions

Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

Universal diophantine equation

1982 ◽  
Vol 47 (3) ◽  
pp. 549-571 ◽  
Author(s):  
James P. Jones
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
Positive Integers ◽  
Martin Davis ◽  
Exponential Relation

In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the formHere P is a polynomial with integer coefficients and the variables range over positive integers.In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved [11] that the exponential relation y = 2x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the formFrom this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets.Now it is well known that the recursively enumerable sets W1, W2, W3, … can be enumerated in such a way that the binary relation x ∈ Wv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,

scholarly journals More on Diophantine Equations

2021 ◽  
Vol 5 (6) ◽  
pp. 26-27
Author(s):  
Sanjay Tahiliani
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Positive Integers

In this paper, we will find the solutions of many Diophantine equations.Some are of the form 2(3 x )+ 5(7y ) +11=z2 for non negative x,y and z. we also investigate solutions ofthe Diophantine equation 2(x+3) +11(3y ) ─ 1= z2 for non negative x,y and z and finally, westudy the Diophantine equations (k!×k)n = (n!×n)k and ( 𝒌! 𝒌 ) 𝒏 = ( 𝒏! 𝒏 ) 𝒌 where k and n are positive integers. We show that the first one holds if and only if k=n and the second one holds if and only if k=n or (k,n) =(1,2),(2,1).We also investigate Diophantine equation u! + v! = uv and u! ─ v! = uv .

Secure Watermarking using Diophantine Equations for Authentication and Recovery

2015 ◽  
Vol 3 (2) ◽  
Author(s):  
Jayashree Nair ◽  
T. Padma
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Jpeg Compression ◽  
Authentication Scheme ◽  
Original Image ◽  
Invariant Features ◽  
Jpeg2000 Compression ◽  
Frequency Properties ◽  
Visual Authentication

This paper describes an authentication scheme that uses Diophantine equations based generation of the secret locations to embed the authentication and recovery watermark in the DWT sub-bands. The security lies in the difficulty of finding a solution to the Diophantine equation. The scheme uses the content invariant features of the image as a self-authenticating watermark and a quantized down sampled approximation of the original image as a recovery watermark for visual authentication, both embedded securely using secret locations generated from solution of the Diophantine equations formed from the PQ sequences. The scheme is mildly robust to Jpeg compression and highly robust to Jpeg2000 compression. The scheme also ensures highly imperceptible watermarked images as the spatio –frequency properties of DWT are utilized to embed the dual watermarks.

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