Ramanujan-Nagell equation

In number theory , the Ramanujan-Nagell equation is an equation of form

${\displaystyle x^{2}+7=2^{n}}$with positive integer solutions and ${\displaystyle x\in \mathbb {N} }$ ${\displaystyle n\in \mathbb {N} }$

Sometimes this equation is also given in the form . ${\displaystyle 2^{n}-7=x^{2}}$

This equation is an example of a Diophantine exponential equation . It was named after the Indian mathematician Srinivasa Ramanujan and the Norwegian mathematician Trygve Nagell .

Solutions to the equation

The only five integer solutions to the equation are: ${\displaystyle x^{2}+7=2^{n}}$

${\displaystyle (x,n)=(1,3)}$, so ${\displaystyle 1^{2}+7=2^{3}}$

${\displaystyle (x,n)=(3,4)}$, so ${\displaystyle 3^{2}+7=2^{4}}$

${\displaystyle (x,n)=(5,5)}$, so ${\displaystyle 5^{2}+7=2^{5}}$

${\displaystyle (x,n)=(11,7)}$, so ${\displaystyle 11^{2}+7=2^{7}}$

${\displaystyle (x,n)=(181,15)}$, so ${\displaystyle 181^{2}+7=2^{15}}$

These five solutions were first mentioned by Ramanujan in 1913. He also suggested that these five solutions are the only integer solutions to this equation. Independently of this, the Norwegian mathematician Wilhelm Ljunggren also came up with this assumption in 1943. A proof of this conjecture but could only Nagell in 1948 supply.

Ramanujan-Nagell numbers

A Mersenne number that is also a triangular number is called a Ramanujan-Nagell number . A Mersenne number is a number of the shape , a triangular number is a number of the shape . If you want to calculate all Mersenne numbers that are also triangular numbers at the same time (also triangular Mersenne numbers ), you have to solve the following equation: ${\displaystyle 2^{b}-1}$ ${\displaystyle {\frac {y\cdot (y+1)}{2}}}$

${\displaystyle 2^{b}-1={\frac {y\cdot (y+1)}{2}}}$

If one reformulates this equation somewhat, one obtains:

${\displaystyle {\begin{aligned}2^{b}-1&=&{\frac {y\cdot (y+1)}{2}}\\8\cdot (2^{b}-1)&=&4\cdot y\cdot (y+1)\\2^{b+3}-8&=&4y^{2}+4y\\2^{b+3}-7&=&4y^{2}+4y+1\\2^{b+3}-7&=&(2y+1)^{2}\end{aligned}}}$

If one now sets and , one obtains the Ramanujan-Nagell equation . Since you already know that this equation only has five solutions , you can calculate the corresponding and : The corresponding Mersenne numbers are therefore: ${\displaystyle n:=b+3}$ ${\displaystyle x:=2y+1}$ ${\displaystyle 2^{n}-7=x^{2}}$ ${\displaystyle (n,x)\in \{(3,1),(4,3),(5,5),(7,11),(15,181)\}}$ ${\displaystyle b=n-3}$ ${\displaystyle y={\frac {x-1}{2}}}$ ${\displaystyle (b,y)\in \{(0,0),(1,1),(2,2),(4,5),(12,90)\}}$ ${\displaystyle 2^{b}-1}$

${\displaystyle 2^{0}-1=0,\quad }$ ${\displaystyle 2^{1}-1=1,\quad }$ ${\displaystyle 2^{2}-1=3,\quad }$ ${\displaystyle 2^{4}-1=15\quad }$ and ${\displaystyle 2^{12}-1=4095}$

The following five Mersenne numbers are also triangular numbers and therefore the only Ramanujan-Nagell numbers:

0, 1, 3, 15, 4095 (sequence A076046 in OEIS )

There are no other Mersenne numbers that are also triangular numbers.

Generalizations

Generalizations of the Ramanujan-Nagell equation have the form

${\displaystyle x^{2}+D=A\cdot B^{n}\quad }$with given integers and variables ${\displaystyle D,A,B\in Z}$ ${\displaystyle x,n}$

They are also called equations of the Ramanujan-Nagell type .

The mathematician Carl Ludwig Siegel was able to show that the number of solutions is finite in all cases. ${\displaystyle (x,n)}$

Example 1:

Be , and . Then the equation is: ${\displaystyle A:=1}$ ${\displaystyle B:=2}$ ${\displaystyle D:=7}$

${\displaystyle x^{2}+7=2^{n}}$

This equation, transformed , results in what is again the original Ramanujan-Nagell equation with the five solutions already mentioned. ${\displaystyle 2^{n}-7=x^{2}}$

Example 2:

Be , and . Then you have to do with the equation . In other words, the equation is: ${\displaystyle A:=4}$ ${\displaystyle B:=2}$ ${\displaystyle D\not =7}$ ${\displaystyle x^{2}+D=4\cdot 2^{n}}$

${\displaystyle x^{2}+D=2^{n+2}}$ With ${\displaystyle D\not =7}$

This type of equation always has at most two solutions. But there are infinitely many for which this equation has exactly two solutions. For example, these two solutions to determine are given here: ${\displaystyle D}$ ${\displaystyle D}$

${\displaystyle (x,n)\in \{(3,5),(45,11)\}}$ For ${\displaystyle D=23}$

${\displaystyle (x,n)\in \{(1,m),(2^{m}-1,2m-1)\}}$for , ${\displaystyle D=2^{m}-1}$ ${\displaystyle m\geq 4}$

Example 3:

Be , and . Then the equation is: ${\displaystyle D:=119}$ ${\displaystyle A:=15}$ ${\displaystyle B:=2}$

${\displaystyle x^{2}+119=15\cdot 2^{n}}$

This equation has the following six solutions:

${\displaystyle (x,n)\in \{(1,3),(11,4),(19,5),(29,6),(61,8),(701,15)\}}$

Lebesgue-Nagell type equations

An equation of form

${\displaystyle x^{2}+D=Ay^{n}\quad }$with given integers and variables ${\displaystyle D,A\in Z}$ ${\displaystyle x,y,n}$

is called the Lebesgue-Nagell type equation . It was named after the French mathematician Victor-Amédée Lebesgue , who could show that the equation

${\displaystyle x^{2}+1=y^{n}}$

has no solution except for the following trivial solutions:

${\displaystyle 0^{2}+1=y^{0}}$, and ${\displaystyle \quad 0^{2}+1=1^{n}}$ ${\displaystyle x^{2}+1=(x^{2}+1)^{1}}$

The latter trivial family of solutions includes, for example, or . ${\displaystyle 1^{2}+1=2^{1}}$ ${\displaystyle 7^{2}+1=50^{1}}$

Example 1:

The two mathematicians Robert Tijdeman and Tarlok Nath Shorey were able to show in 1986 that the number of solutions to the equation is finite in every case. ${\displaystyle x^{2}+D=Ay^{n}}$

Example 2:

The three mathematicians Yann Bugeaud, Maurice Mignotte and Samir Siksek solved equations of this type for and in 2006 . ${\displaystyle A=1}$ ${\displaystyle 1\leq D\leq 100}$

In particular, they were able to show that the generalization of the Ramanujan-Nagell equation

${\displaystyle x^{2}+7=y^{n}}$ With ${\displaystyle n\geq 3}$

only has the five positive integer solutions . ${\displaystyle (x,y,n)\in \{(1,2,3),(3,2,4),(5,2,5),(11,2,7),(181,2,15)\}}$

For this equation still has the trivial solution , that is . ${\displaystyle 0\leq n\leq 2}$ ${\displaystyle 0^{2}+7=7^{1}}$ ${\displaystyle (x,y,n)\in \{(0,7,1)\}}$

Solutions of this equation such as or can be converted to or , which in turn leads to the already known solutions . ${\displaystyle 1^{2}+7=8^{1}}$ ${\displaystyle 3^{2}+7=4^{2}}$ ${\displaystyle 1^{2}+7=2^{3}}$ ${\displaystyle 3^{2}+7=2^{4}}$ ${\displaystyle (x,y,n)\in \{(1,2,3),(3,2,4)\}}$

See also

• Pillai conjecture : with only has a finite number of solutions. ${\displaystyle Ax^{n}-By^{m}=C}$ ${\displaystyle A,B,C\in \mathbb {N} }$

Web links

• Kalyan Chakraborty, Azizul Hoque, Richa Sharma: Complete solutions of certain Lebesgue-Ramanujan-Nagell Type equations. September 28, 2019, pp. 1–16 , accessed on January 11, 2020 . 
• Fadwa S. Abu Muriefah, Yann Bugeaud: The Diophantine equation x ² + c = y ⁿ  : a brief overview. Revista Colombiana de Matemáticas 40 , 2006, pp. 31-37 , accessed on January 11, 2020 . 

Individual evidence

1. ↑ Srinivasa Ramanujan : Question 446 , J. Indian Math. Soc. 5 (1913), 120, Collected papers, Cambridge University Press (1927), p. 327
2. ↑ Wilhelm Ljunggren : Oppgave nr 2 ., Norske Mat Tidsskrift 25 (1943), p 29 (Norwegian)
3. ↑ ^{a b} N. Saradha, Anitha Srinivasan: Generalized Lebesgue-Ramanujan-Nagell Equations. Diophantine Equations, 2008, pp. 207-223 , accessed January 11, 2020 . 
4. ↑ Attila Bérczes, István Pink: On generalized Lebesgue-Ramanujan-Nagell equations. Analele Universitatii Ovidius Constanta, Seria Matematica 22 (1), January 10, 2014, pp. 51-71 , accessed on January 11, 2020 . 
5. ↑ Trygve Nagell : The Diophantine equation x ² + 7 = 2 ⁿ . Ark. Math. 4 (13), January 13, 1960, pp. 185-187 , accessed January 11, 2020 . 
6. ↑ Carl Ludwig Siegel : Approximation of algebraic numbers. Sentence 7 on p. 204. Math. Time. 10 , 1921, pp. 173-213 , accessed January 11, 2020 . 
7. ↑ Roger Apéry : Sur une èquation diophantienne , CR Acad. Sci. Paris Sér. A 251 (1960), pp. 1263–1264 and pp. 1451–1452 (French)
8. ↑ N. Saradha, Anitha Srinivasan: Generalized Lebesgue-Ramanujan-Nagell Equations. Section (2.3) on p. 2 (= p. 208). Diophantine Equations, 2008, pp. 207-223 , accessed January 11, 2020 . 
9. ↑ N. Saradha, Anitha Srinivasan: Generalized Lebesgue-Ramanujan-Nagell Equations. Statement before Proposition 2.1, p. 4f. Diophantine Equations, 2008, pp. 207-223 , accessed January 11, 2020 . 
10. ↑ Victor-Amédée Lebesgue : Sur l'impossibilité, en nombres entiers, de l'équation x ^m = y ² +1. Nouvelles Annales de Mathématiques (1) 9 , 1850, pp. 178-181 , accessed on January 11, 2020 (French). 
11. ↑ Tarlok Nath Shorey, Robert Tijdeman : Exponential Diophantine equations , Theorem 10.6, Cambridge Tracts in Mathematics 87 , Cambridge University Press, Cambridge, 1986
12. ↑ Yann Bugeaud, Maurice Mignotte, Samir Siksek: . Classical and modular Approaches to exponential Diophantine equations and II The Lebesgue Nagell equation. Theorem 1. Compos. Math. 142 (1), January 2006, pp. 31-62 , accessed January 11, 2020 .