CYCLIC SYSTEMS OF SIMULTANEOUS CONGRUENCES

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 &sube; {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 &sube; N has a finite-fold Diophantine representation, then M is computable.

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A New Proof of Smoryński’s Theorem

10.20944/preprints201709.0127.v2 ◽

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.

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

10.20944/preprints201804.0121.v1 ◽

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 &sube; {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 &sube; N has a finite-fold Diophantine representation, then M is computable.

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A New Proof of Smoryński’s Theorem

10.20944/preprints201709.0127.v1 ◽

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.

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A GENERALIZATION OF A THEOREM OF BUMBY ON QUARTIC DIOPHANTINE EQUATIONS

International Journal of Number Theory ◽

10.1142/s1793042106000474 ◽

2006 ◽

Vol 02 (02) ◽

pp. 195-206 ◽

Cited By ~ 6

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).

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

International Journal of Number Theory ◽

10.1142/s1793042119500593 ◽

2019 ◽

Vol 15 (05) ◽

pp. 1069-1074 ◽

Cited By ~ 2

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].

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The Diophantine equation Fn = P(x)

International Journal of Number Theory ◽

10.1142/s1793042120501079 ◽

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].

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

International Journal of Number Theory ◽

10.1142/s1793042121500731 ◽

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]

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Fermat's Last Theorem: A Proof by Contradiction

10.14293/s2199-1006.1.sor-.ppnryjc.v1 ◽

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.

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

Glasnik Matematicki ◽

10.3336/gm.55.2.03 ◽

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).

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