Displaying similar documents to “Complete solutions of a Lebesgue-Ramanujan-Nagell type equation”

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.

Lucas sequences and repdigits

Hayder Raheem Hashim, Szabolcs Tengely (2022)

Mathematica Bohemica

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Let ( G n ) n 1 be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are { U n } and { V n } , respectively. We show that the Diophantine equation G n = B · ( g l m - 1 ) / ( g l - 1 ) has only finitely many solutions in n , m + , where g 2 , l is even and 1 B g l - 1 . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral...

Integral points on the elliptic curve y 2 = x 3 - 4 p 2 x

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

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Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E : y 2 = x 3 - 4 p 2 x . Further, let N ( p ) denote the number of pairs of integral points ( x , ± y ) on E with y > 0 . We prove that if p 17 , then N ( p ) 4 or 1 depending on whether p 1 ( mod 8 ) or p - 1 ( mod 8 ) .

The Mordell-Weil bases for the elliptic curve y 2 = x 3 - m 2 x + m 2

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

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Let D m be an elliptic curve over of the form y 2 = x 3 - m 2 x + m 2 , where m is an integer. In this paper we prove that the two points P - 1 = ( - m , m ) and P 0 = ( 0 , m ) on D m can be extended to a basis for D m ( ) under certain conditions described explicitly.

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 .

The number of solutions to the generalized Pillai equation ± r a x ± s b y = c .

Reese Scott, Robert Styer (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider N , the number of solutions ( x , y , u , v ) to the equation ( - 1 ) u r a x + ( - 1 ) v s b y = c in nonnegative integers x , y and integers u , v { 0 , 1 } , for given integers a > 1 , b > 1 , c > 0 , r > 0 and s > 0 . When gcd ( r a , s b ) = 1 , we show that N 3 except for a finite number of cases all of which satisfy max ( a , b , r , s , x , y ) < 2 · 10 15 for each solution; when gcd ( a , b ) > 1 , we show that N 3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving N = 3 solutions.

A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

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We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system n = a 1 + a 2 + + a s - 1 , a 1 a 2 a s - 1 ( a 1 + a 2 + + a s - 1 ) = b s has positive integer or rational solutions n , b , a i , i = 1 , 2 , , s - 1 , s 3 . Using the theory of elliptic curves, we prove that it has no positive integer solution for s = 3 , but there are infinitely many positive integers n such that it has a positive integer solution for s 4 . As a corollary, for s 4 and any positive integer n , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions...

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.

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 .

On invariants of elliptic curves on average

Amir Akbary, Adam Tyler Felix (2015)

Acta Arithmetica

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We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let e E ( p ) be the exponent of the group of rational points of the reduction modulo p of E over the finite field p . Let be the family of elliptic curves E a , b : y 2 = x 3 + a x + b , where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, 1 / | | E p x e E k ( p ) = C k l i ( x k + 1 ) + O ( ( x k + 1 ) / ( l o g x ) c ) as x → ∞, as long...

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 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.