Contributions to the theory of diophantine equations I. On the representation of integers by binary forms

1968 ◽  
Vol 263 (1139) ◽  
pp. 173-191 ◽  
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Binary Form ◽  
Effective Algorithm ◽  
Algebraic Numbers ◽  
Binary Forms ◽  
Integer Coefficients ◽  
Representation Of Integers

An effective algorithm is established for solving in integers x, y any Diophantine equation of the type/( x, y ) = m , where/ denotes an irreducible binary form with integer coefficients and degree at least 3. The magnitude, relative to m, of the bound furnished by the algorithm for the size of all the solutions of the equation is investigated, and, in consequence, there is obtained the first generally effective improvement on the well known result of Liouville (1844) concerning the accuracy with which algebraic numbers can be approximated by rationals.

Contributions to the theory of Diophantine equations II. The Diophantine equation y 2 = x 3 + k

1968 ◽  
Vol 263 (1139) ◽  
pp. 193-208 ◽  
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Binary Form ◽  
Effective Algorithm ◽  
Integer Coefficients ◽  
Explicit Bound

This paper is a sequel to Part I (Baker 1968) in which an effective algorithm was established for solving in integers x, y any Diophantine equation of the type y) = m, where ^denotes an irreducible binary form with integer coefficients and degree at least 3. Here the algorithm is utilized to obtain an explicit bound, free from unknown constants, for the size of all the solutions of the equation. As a consequence of the cubic case of the result, it is proved that, for any integer 4= 9, all integers x, y satisfying the equation of the title have absolute values at most exp { (10101 A:|)10 }.

scholarly journals On totally reducible binary forms: II.

2002 ◽  
Vol Volume 25 ◽  
Author(s):  
C Hooley
Keyword(s):  
Asymptotic Formula ◽  
Principal Result ◽  
Binary Form ◽  
Binary Forms ◽  
Linear Forms ◽  
Integer Coefficients ◽  
International Audience ◽  
Product Of Linear Forms ◽  
Positive Integers

International audience Let $f$ be a binary form of degree $l\geq3$, that is, a product of linear forms with integer coefficients. The principal result of this paper is an asymptotic formula of the shape $n^{2/l}(C(f)+O(n^{-\eta_l+\varepsilon}))$ for the number of positive integers not exceeding $n$ that are representable by $f$; here $C(f)>0$ and $\eta_l>0$.

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,

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.

scholarly journals Quadratic ideals, indefinite quadratic forms and some specific diophantine equations

2018 ◽  
Vol 36 (3) ◽  
pp. 173-192
Author(s):  
Ahmet Tekcan ◽  
Seyma Kutlu
Keyword(s):  
Diophantine Equation ◽  
Quadratic Forms ◽  
Diophantine Equations ◽  
Algebraic Properties ◽  
Integer Solutions ◽  
Indefinite Quadratic

Let $k\geq 1$ be an integer and let $P=k+2,Q=k$ and $D=k^{2}+4$. In this paper, we derived some algebraic properties of quadratic ideals $I_{\gamma}$ and indefinite quadratic forms $F_{\gamma }$ for quadratic irrationals $\gamma$, and then we determine the set of all integer solutions of the Diophantine equation $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

Triple Binary Forms; the complete system for a single (1, 1, 1) form with its geometrical interpretation

1925 ◽  
Vol 22 (5) ◽  
pp. 688-693 ◽  
Author(s):  
W. Saddler
Keyword(s):  
Complete System ◽  
Geometrical Interpretation ◽  
Binary Form ◽  
Binary Forms ◽  
General Terms ◽  
American Math

In the Transactions of the American Math. Society, vol. I, 1900, and vol. IV, 1904, Mr E. Kasner treats exhaustively the (2, 2) double binary form and discusses the theory connected with double binary forms and multiple binary forms in general terms. He shows the relations between the systems of multiple binary forms with digredient variables and the forms with cogredient variables. Hitherto nothing seems to have been written on systems of triple binary forms (with regard to higher forms see a paper on the (1, 1, 1, 1) form by C. Segre); so here I propose to discuss the complete system of a (1, 1, 1) binary form which consists of six forms connected by one syzygy. When two of the variables are the same we naturally get the (2, 1) form and when the three are the same We get the (3) or cubic binary form.

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 Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

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.

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