Ramanujan–Nagell Equation
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In mathematics, in the field of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, the Ramanujan–Nagell equation is an equation between a
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
and a number that is seven less than a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negati ...
. It is an example of an
exponential Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
, an equation to be solved in integers where one of the variables appears as an
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
. The equation is named after
Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
, who conjectured that it has only five integer solutions, and after
Trygve Nagell Trygve Nagell or Trygve Nagel (July 13, 1895 in Oslo – January 24, 1988 in Uppsala) was a Norwegian mathematician, known for his works on Diophantine equations in number theory. Education and career He was born Nagel and adopted the sp ...
, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum
Hamming distance In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chang ...
5 or 6.


Equation and solution

The equation is :2^n-7=x^2 \, and solutions in natural numbers ''n'' and ''x'' exist just when ''n'' = 3, 4, 5, 7 and 15 . This was conjectured in 1913 by Indian mathematician
Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician
Trygve Nagell Trygve Nagell or Trygve Nagel (July 13, 1895 in Oslo – January 24, 1988 in Uppsala) was a Norwegian mathematician, known for his works on Diophantine equations in number theory. Education and career He was born Nagel and adopted the sp ...
. The values of ''n'' correspond to the values of ''x'' as:- :''x'' = 1, 3, 5, 11 and 181 .


Triangular Mersenne numbers

The problem of finding all numbers of the form 2''b'' − 1 (
Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
s) which are
triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, ...
is equivalent: : \begin & \ 2^b-1 = \frac \\ pt\Longleftrightarrow & \ 8(2^b-1) = 4y(y+1) \\ \Longleftrightarrow & \ 2^-8 = 4y^2+4y \\ \Longleftrightarrow & \ 2^-7 = 4y^2+4y+1 \\ \Longleftrightarrow & \ 2^-7 = (2y+1)^2 \end The values of ''b'' are just those of ''n'' − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan–Nagell numbers) are: :\frac = \frac for ''x'' = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more .


Equations of Ramanujan–Nagell type

An equation of the form : x^2 + D = A B^n for fixed ''D'', ''A'' , ''B'' and variable ''x'', ''n'' is said to be of ''Ramanujan–Nagell type''. The result of
Siegel Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). ...
implies that the number of solutions in each case is finite. By representing n = 3m + r with r\in\ and B^n = B^r y^3 with y=B^m, the equation of Ramanujan–Nagell type is reduced to three
Mordell curve In algebra, a Mordell curve is an elliptic curve of the form ''y''2 = ''x''3 + ''n'', where ''n'' is a fixed non-zero integer. These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He sh ...
s (indexed by r), each of which has a finite number of integer solutions: : r=0:\qquad (Ax)^2 = (Ay)^3 - A^2D, : r=1:\qquad (ABx)^2 = (ABy)^3 - A^2B^2D, : r=2:\qquad (AB^2x)^2 = (AB^2y)^3 - A^2B^4D. The equation with A=1,\ B=2 has at most two solutions, except in the case D=7 corresponding to the Ramanujan–Nagell equation. There are infinitely many values of ''D'' for which there are two solutions, including D = 2^m - 1.


Equations of Lebesgue–Nagell type

An equation of the form : x^2 + D = A y^n for fixed ''D'', ''A'' and variable ''x'', ''y'', ''n'' is said to be of ''Lebesgue–Nagell type''. This is named after Victor-Amédée Lebesgue, who proved that the equation : x^2 + 1 = y^n has no nontrivial solutions. Results of Shorey and Tijdeman imply that the number of solutions in each case is finite. Bugeaud, Mignotte and Siksek solved equations of this type with ''A'' = 1 and 1 ≤ ''D'' ≤ 100. In particular, the following generalization of the Ramanujan-Nagell equation: :y^n-7=x^2 \, has positive integer solutions only when ''x'' = 1, 3, 5, 11, or 181.


See also

* Pillai's conjecture * Scientific equations named after people


Notes


References

* * * * * * * * *


External links

*
Can ''N''2 + ''N'' + 2 Be A Power Of 2?
Math Forum discussion {{DEFAULTSORT:Ramanujan-Nagell equation Diophantine equations Srinivasa Ramanujan