In mathematics, in the field of number theory, the RamanujanâNagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.
The equation is
and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15 (sequence A060728 in the OEIS).
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of n correspond to the values of x as:-
The problem of finding all numbers of the form 2b − 1 ( Mersenne numbers) which are triangular is equivalent:
The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as RamanujanâNagell numbers) are:
for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more (sequence A076046 in the OEIS).
An equation of the form
for fixed D, A, B and variable x, n is said to be of RamanujanâNagell type. The result of Siegel [2] implies that the number of solutions in each case is finite. [3] By representing with and with , the equation of RamanujanâNagell type is reduced to three Mordell curves (indexed by ), each of which has a finite number of integer solutions:
The equation with has at most two solutions, except in the case corresponding to the RamanujanâNagell equation. There are infinitely many values of D for which there are two solutions, including . [1]
An equation of the form
for fixed D, A and variable x, y, n is said to be of LebesgueâNagell type. This is named after Victor-AmĂ©dĂ©e Lebesgue, who proved that the equation
has no nontrivial solutions. [4]
Results of Shorey and Tijdeman [5] imply that the number of solutions in each case is finite. [6] Bugeaud, Mignotte and Siksek [7] solved equations of this type with A = 1 and 1 ≤ D ≤ 100. In particular, the following generalization of the RamanujanâNagell equation:
has positive integer solutions only when x = 1, 3, 5, 11, or 181.
In mathematics, in the field of number theory, the RamanujanâNagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.
The equation is
and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15 (sequence A060728 in the OEIS).
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of n correspond to the values of x as:-
The problem of finding all numbers of the form 2b − 1 ( Mersenne numbers) which are triangular is equivalent:
The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as RamanujanâNagell numbers) are:
for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more (sequence A076046 in the OEIS).
An equation of the form
for fixed D, A, B and variable x, n is said to be of RamanujanâNagell type. The result of Siegel [2] implies that the number of solutions in each case is finite. [3] By representing with and with , the equation of RamanujanâNagell type is reduced to three Mordell curves (indexed by ), each of which has a finite number of integer solutions:
The equation with has at most two solutions, except in the case corresponding to the RamanujanâNagell equation. There are infinitely many values of D for which there are two solutions, including . [1]
An equation of the form
for fixed D, A and variable x, y, n is said to be of LebesgueâNagell type. This is named after Victor-AmĂ©dĂ©e Lebesgue, who proved that the equation
has no nontrivial solutions. [4]
Results of Shorey and Tijdeman [5] imply that the number of solutions in each case is finite. [6] Bugeaud, Mignotte and Siksek [7] solved equations of this type with A = 1 and 1 ≤ D ≤ 100. In particular, the following generalization of the RamanujanâNagell equation:
has positive integer solutions only when x = 1, 3, 5, 11, or 181.