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A modified Ramanujan–Nagell equation

In the field of number theory, the Ramanujan–Nagell 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.

It is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture.

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, 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:- n = 3, 4, 5, 7 and 15 x = 1, 3, 5, 11 and 181.

Similarly what can we say about the equation:

2^n - 3 = x^3 ?

How many solutions does it have ?

I can only think of 2 solutions : ( n = 2 and x = 1 ) and (n = 7 = x = 5)

How can we prove that there are only two solutions to the above equation ?

Can any one come up with a 3rd solution ?

Note by Vijay Simha
1 year ago

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It looks like having no further solution. But I am not enough strong to prove that. U can think of programming this out. Then for large value u will have searches. Shyambhu Mukherjee · 1 year ago

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