The problem
The Diophantine equation \(x^2 + 7 = 2^n\) has exactly five integer solutions: \((x,n) \in \{(1,3),(3,4),(5,5),(11,7),(181,15)\}\). (Resolved 1948.)
The Ramanujan–Nagell equation
historicdiophantine-equations exponential-diophantine
History & significance
Ramanujan conjectured in 1913 that \(x^2 + 7 = 2^n\) has exactly the solutions \((x,n) = (1,3),(3,4),(5,5),(11,7),(181,15)\); Nagell proved completeness in 1948 using algebraic number theory in \(\mathbb{Q}(\sqrt{-7})\). It became the prototype for Lebesgue–Nagell-type exponential equations, whose modern treatment (Bugeaud–Mignotte–Siksek) blends lower bounds for linear forms in logarithms with modular methods.
Connected problems
More in Diophantine equations
References
The resolution (human proof)
Nagell (1948) proved Ramanujan's list complete: unique factorization in the ring of integers of Q(√−7) forces the five known solutions of x² + 7 = 2ⁿ to be the only ones.
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