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Ramanujan–Nagell equation - mklabgr - 07-26-2026
Ramanujan–Nagell equation
Summary
The Ramanujan–Nagell equation is an exponential Diophantine equation given by $2^n - 7 = x^2$, which asks for integer solutions where a power of two minus seven equals a perfect square. First conjectured in 1913 by Indian mathematician Srinivasa Ramanujan and definitively proved in 1948 by Norwegian mathematician Trygve Nagell,
The equation is famous for having only five solutions for $n$ ($n = 3, 4, 5, 7,$ and $15$). Beyond its interest in number theory—where it is equivalent to finding Mersenne numbers that are also triangular—the equation plays a crucial role in coding theory by proving the non-existence of certain perfect binary error-correcting codes.
ARTICLE