scholarly journals A New Proof of Smoryński’s Theorem

2017 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Diophantine Equations ◽  
Recursively Enumerable ◽  
Integer Solutions ◽  
Positive Integers

We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable. Analogous theorems hold for Diophantine equations D(x1,…, xp) = 0, where p ∈ N \ {0} and for every i ∈ {1,…, p} the polynomial D(x1,…, xp) involves a monomial M with a non-zero coeffcient such that xi divides M.

scholarly journals A New Proof of Smoryński’s Theorem

2017 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Diophantine Equations ◽  
Recursively Enumerable ◽  
Integer Solutions ◽  
Positive Integers

We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable, (4) analogous theorems hold for Diophantine equations D(x1, …, xp) = 0, where p ∈ N\{0} and for every i ∈ {1, …, p} the polynomial D(x1, …, xp) involves a monomial M with a non-zero coefficient such that xi divides M, (5) the set of all Diophantine equations which have at most k variables (where k ≥ 9) and at most finitely many solutions in non-negative integers is not recursively enumerable.

scholarly journals CYCLIC SYSTEMS OF SIMULTANEOUS CONGRUENCES

2010 ◽  
Vol 06 (02) ◽  
pp. 219-245 ◽  
Author(s):  
JEFFREY C. LAGARIAS
Keyword(s):  
Diophantine Equations ◽  
Extremal Solutions ◽  
Infinitely Many Solutions ◽  
Cyclic System ◽  
Positivity Condition ◽  
The Family ◽  
Integer Solutions ◽  
Almost All ◽  
Positive Integers ◽  
Simultaneous Congruences

This paper considers the cyclic system of n ≥ 2 simultaneous congruences [Formula: see text] for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers qi ≥ 2, with gcd (q1q2 ⋯ qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807,…. If the positivity condition on the integers qi is dropped, then for r = 1 these systems of congruences, taken ( mod |qi|), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equations [Formula: see text] depending on three parameters (r, s, m).

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).

Complete solutions of the simultaneous Pell’s equations x2 − (a2 − 1)y2 = 1 and y2 − pz2 = 1

2019 ◽  
Vol 15 (05) ◽  
pp. 1069-1074 ◽  
Author(s):  
Hai Yang ◽  
Ruiqin Fu
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Integer Solution ◽  
System Of Equations ◽  
Positive Integer Solution ◽  
Elementary Methods ◽  
Integer Solutions ◽  
Positive Integers ◽  
Complete Solutions

Let [Formula: see text] be a positive integer with [Formula: see text], and let [Formula: see text] be an odd prime. In this paper, by using certain properties of Pell’s equations and quartic diophantine equations with some elementary methods, we prove that the system of equations [Formula: see text] [Formula: see text] and [Formula: see text] has positive integer solutions [Formula: see text] if and only if [Formula: see text] and [Formula: see text] satisfy [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are positive integers. Further, if the above condition is satisfied, then [Formula: see text] has only the positive integer solution [Formula: see text]. By the above result, we can obtain the following corollaries immediately. (i) If [Formula: see text] or [Formula: see text], then [Formula: see text] has no positive integer solutions [Formula: see text]. (ii) For [Formula: see text], [Formula: see text] has only the positive integer solutions [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

On the solvability of the simultaneous Pell equations x2 − ay2 = 1 and y2 − bz2 = v12

2021 ◽  
pp. 1-12
Author(s):  
Ruiqin Fu ◽  
Hai Yang
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Prime Divisors ◽  
Positive Integer Solution ◽  
Pell Equations ◽  
Integer Solutions ◽  
Least Solution ◽  
Simultaneous Pell Equations ◽  
Necessary And Sufficient ◽  
Positive Integers

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

scholarly journals Fermat's Last Theorem: A Proof by Contradiction

2021 ◽  
Author(s):  
Benson Schaeffer
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Algebraic Proof ◽  
Fermat's Last Theorem ◽  
Binomial Expansion ◽  
Proof By Contradiction ◽  
Fermat’S Last Theorem ◽  
Integer Solutions ◽  
Positive Integers ◽  
Fermat's Equation

In this paper I offer an algebraic proof by contradiction of Fermat’s Last Theorem. Using an alternative to the standard binomial expansion, (a+b) n = a n + b Pn i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I show that a simple rewrite of the Fermat’s equation stating the theorem, A p + B p = (A + B − D) p , A, B, D and p positive integers, D < A < B, p ≥ 3 and prime, entails the contradiction, A(B − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 A i−1−j (A + B − D) j−1 # + B(A − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 B i−1−j (A + B − D) j−1 # = 0, the sum of two positive integers equal to zero. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat’s Last Theorem. AMS 2020 subject classification: 11A99, 11D41 Diophantine equations, Fermat’s equation ∗The corresponding author. E-mail: [email protected] 1 1 Introduction To prove Fermat’s Last Theorem, it suffices to show that the equation A p + B p = C p (1In this paper I offer an algebraic proof by contradiction of Fermat’s Last Theorem. Using an alternative to the standard binomial expansion, (a+b) n = a n + b Pn i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I show that a simple rewrite of the Fermat’s equation stating the theorem, A p + B p = (A + B − D) p , A, B, D and p positive integers, D < A < B, p ≥ 3 and prime, entails the contradiction, A(B − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 A i−1−j (A + B − D) j−1 # + B(A − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 B i−1−j (A + B − D) j−1 # = 0, the sum of two positive integers equal to zero. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat’s Last Theorem.

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 Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

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