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**What is Ramanujan-Nagell equation?**

The Ramanujan-Nagell equation is a Diophantine equation, which is an equation to be solved in integers where one of the variables appears as an exponent. It was named after Srinivasa Ramanujan, who conjectured there would be only five integer solutions, and after Trygve Nagell, who proved it.

The equation is as follows:

2^{n} – 7 = x^{2}

where n and x are integers.

The only five integer solutions to the equation are:

(n, x) = (3, 1), (4, 3), (5, 5), (7, 11), and (15, 181)

**Uses of Ramanujan-Nagell equation**

The Ramanujan-Nagell equation has no direct applications. However, it is a good example of a Diophantine equation with a very limited number of solutions. This makes it a good testing ground for new methods of solving Diophantine equations.

**Why study Ramanujan-Nagell equation?**

Studying the Ramanujan-Nagell equation gives a strong foundation in number theory and Diophantine equations. This can be useful in a number of careers, including mathematician, computer scientist (cryptography and coding theory), physicist (quantum mechanics and string theory), and financial analyst (risk assessment and portfolio optimization).

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