The problem
The only consecutive powers among integers above 1 are \(8\) and \(9\): the equation \(x^p - y^q = 1\) with \(x,y > 0\), \(p,q > 1\) has the unique solution \(3^2 - 2^3 = 1\). (Resolved 2002.)
Catalan's conjecture (Mihăilescu's theorem)
historicdiophantine-equations exponential-diophantine
History & significance
Catalan asked in 1844 whether 8 and 9 are the only consecutive powers. Lebesgue settled exponent 2 vs odd primes (1850); Tijdeman (1976) used Baker's theory of linear forms in logarithms to bound the exponents effectively — the first general finiteness with explicit bounds. Mihăilescu (2002, published 2004) closed it completely with a strikingly short argument built on cyclotomic fields and Lifting-the-Exponent, a proof short enough to teach.
Connected problems
- related Pillai's Conjecture
- related The Ramanujan–Nagell equation
More in Diophantine equations
- same area The Ramanujan–Nagell equation
References
The resolution (human proof)
Mihăilescu (2002) proved the only consecutive powers are 8 and 9, using cyclotomic-field machinery and LTE-type arguments to force a contradiction from any hypothetical second solution — after Tijdeman (1976) had made the exponents effectively bounded.
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