Displaying similar documents to “On some Diophantine equations involving balancing numbers”

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

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We study the Diophantine equations ( k ! ) n - k n = ( n ! ) k - n k and ( k ! ) n + k n = ( n ! ) k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or ( k , n ) = ( 1 , 2 ) , ( 2 , 1 ) and that the second one holds if and only if k = n .

Finiteness results for Diophantine triples with repdigit values

Attila Bérczes, Florian Luca, István Pink, Volker Ziegler (2016)

Acta Arithmetica

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Let g ≥ 2 be an integer and g be the set of repdigits in base g. Let g be the set of Diophantine triples with values in g ; that is, g is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set g . We prove effective finiteness results for the set g .

A note on the article by F. Luca “On the system of Diophantine equations a ² + b ² = ( m ² + 1 ) r and a x + b y = ( m ² + 1 ) z ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

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Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying A + B ( - 1 ) = ( m + ( - 1 ) ) r . We prove that the Diophantine equation | A | x + | B | y = ( m ² + 1 ) z has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever r > 10 74 or m > 10 34 . Our result is an explicit refinement of a theorem due to F. Luca.

On the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2

Ruizhou Tong (2021)

Czechoslovak Mathematical Journal

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Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 x , p ± 3 ( mod 8 ) , the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution except when p = 3 or p is of the form p = 2 a 0 2 + 1 , where a 0 > 1 is an odd positive integer. (2) if 2 x , 2 y , y 2 , 4 , then the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution.

A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque (2024)

Czechoslovak Mathematical Journal

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S. S. Pillai proved that for a fixed positive integer a , the exponential Diophantine equation x y - y x = a , min ( x , y ) > 1 , has only finitely many solutions in integers x and y . We prove that when a is of the form 2 z 2 , the above equation has no solution in integers x and y with gcd ( x , y ) = 1 .

On the Diophantine equation j = 1 k j F j p = F n q

Gökhan Soydan, László Németh, László Szalay (2018)

Archivum Mathematicum

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Let F n denote the n t h term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F 1 p + 2 F 2 p + + k F k p = F n q in the positive integers k and n , where p and q are given positive integers. A complete solution is given if the exponents are included in the set { 1 , 2 } . Based on the specific cases we could solve, and a computer search with p , q , k 100 we conjecture that beside the trivial solutions only F 8 = F 1 + 2 F 2 + 3 F 3 + 4 F 4 , F 4 2 = F 1 + 2 F 2 + 3 F 3 , and F 4 3 = F 1 3 + 2 F 2 3 + 3 F 3 3 satisfy the title equation.

Complete solution of the Diophantine equation x y + y x = z z

Mihai Cipu (2019)

Czechoslovak Mathematical Journal

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The triples ( x , y , z ) = ( 1 , z z - 1 , z ) , ( x , y , z ) = ( z z - 1 , 1 , z ) , where z , satisfy the equation x y + y x = z z . In this paper it is shown that the same equation has no integer solution with min { x , y , z } > 1 , thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

On systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

Colloquium Mathematicae

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We consider systems of equations of the form x i + x j = x k and x i · x j = x k , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

Mersenne numbers as a difference of two Lucas numbers

Murat Alan (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let ( L n ) n 0 be the Lucas sequence. We show that the Diophantine equation L n - L m = M k has only the nonnegative integer solutions ( n , m , k ) = ( 2 , 0 , 1 ) , ( 3 , 1 , 2 ) , ( 3 , 2 , 1 ) , ( 4 , 3 , 2 ) , ( 5 , 3 , 3 ) , ( 6 , 2 , 4 ) , ( 6 , 5 , 3 ) where M k = 2 k - 1 is the k th Mersenne number and n > m .

Solutions of the Diophantine Equation 7 X 2 + Y 7 = Z 2 from Recurrence Sequences

Hayder R. Hashim (2020)

Communications in Mathematics

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Consider the system x 2 - a y 2 = b , P ( x , y ) = z 2 , where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7 X 2 + Y 7 = Z 2 if ( X , Y ) = ( L n , F n ) (or ( X , Y ) = ( F n , L n ) ) where { F n } and { L n } represent the sequences of Fibonacci numbers and Lucas numbers...

A diophantine equation involving special prime numbers

Stoyan Dimitrov (2023)

Czechoslovak Mathematical Journal

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Let [ · ] be the floor function. In this paper, we prove by asymptotic formula that when 1 < c < 3441 2539 , then every sufficiently large positive integer N can be represented in the form N = [ p 1 c ] + [ p 2 c ] + [ p 3 c ] + [ p 4 c ] + [ p 5 c ] , where p 1 , p 2 , p 3 , p 4 , p 5 are primes such that p 1 = x 2 + y 2 + 1 .

On the diophantine equation x 2 + 2 a 3 b 73 c = y n

Murat Alan, Mustafa Aydin (2023)

Archivum Mathematicum

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In this paper, we find all integer solutions ( x , y , n , a , b , c ) of the equation in the title for non-negative integers a , b and c under the condition that the integers x and y are relatively prime and n 3 . The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.

On x n + y n = n ! z n

Susil Kumar Jena (2018)

Communications in Mathematics

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In p. 219 of R.K. Guy’s , 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation x n + y n = n ! z n has no integer solutions with n + and n > 2 . But, contrary to this expectation, we show that for n = 3 , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition gcd ( x , y , z ) = 1 .

The exceptional set for Diophantine inequality with unlike powers of prime variables

Wenxu Ge, Feng Zhao (2018)

Czechoslovak Mathematical Journal

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Suppose that λ 1 , λ 2 , λ 3 , λ 4 are nonzero real numbers, not all negative, δ > 0 , 𝒱 is a well-spaced set, and the ratio λ 1 / λ 2 is algebraic and irrational. Denote by E ( 𝒱 , N , δ ) the number of v 𝒱 with v N such that the inequality | λ 1 p 1 2 + λ 2 p 2 3 + λ 3 p 3 4 + λ 4 p 4 5 - v | < v - δ has no solution in primes p 1 , p 2 , p 3 , p 4 . We show that E ( 𝒱 , N , δ ) N 1 + 2 δ - 1 / 72 + ε for any ε > 0 .