Definition:Ramanujan-Nagell Equation

Definition

The Ramanujan-Nagell equation is the Diophantine equation:

$x^2 + 7 = 2^n$

Also known as

The Ramanujan-Nagell equation is also known as Ramanujan's equation and Ramanujan's square equation.

Also see

• Five Ramanujan-Nagell Numbers
• Solutions of Ramanujan-Nagell Equation
• Results about the Ramanujan-Nagell equation can be found here.

Source of Name

This entry was named for Srinivasa Ramanujan and Trygve Nagell.

Historical Note

Srinivasa Ramanujan conjectured in $1913$ that the equation $x^2 + 7 = 2^n$ had only $5$ integer solutions.

The same conjecture was made independently in $1943$ by Wilhelm Ljunggren.

The conjecture was proved in $1948$ by Trygve Nagell

Sources

• 1913: Srinivasa Ramanujan: Question 464 (J. Ind. Math. Soc Vol. 5: p. 130)
• 1943: Wilhelm Ljunggren: Oppgave nr 2 (Norsk Mat. Tidsskr Vol. 25: p. 29)
• 1948: Trygve Nagell: Løsning till oppgave nr 2 (Norsk Mat. Tidsskr Vol. 30: pp. 62 – 64)
• 1960: Trygve Nagell: The Diophantine Equation $x^2 + 7 = 2^n$ (Arkiv för Matematik Vol. 4: pp. 185 – 187)
• 1987: Wells Johnson: The Diophantine Equation $X^2 + 7 = 2^n$ (Amer. Math. Monthly Vol. 94: pp. 59 – 62)  www.jstor.org/stable/2323504

• Weisstein, Eric W. "Ramanujan's Square Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujansSquareEquation.html