Tijdeman's theorem

For each fixed integer greater than one, only finitely many pairs of perfect powers differ by that integer. It follows from lower bounds for linear forms in logarithms.

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Linear forms in logarithms: Expressions formed from integer combinations of logarithms of algebraic numbers, whose nonzero values admit quantitative lower bounds. Tijdeman's proof uses such lower bounds to control the exponents in nearby powers.
Robert Tijdeman: Dutch mathematician known for work in number theory, especially Diophantine equations and transcendence methods. He proved the finiteness result that bears his name.
Effective bound: A numerical bound that can, in principle, be calculated from the stated data. Tijdeman's methods give bounds on possible exponents, though the resulting bounds are enormous.
Integer: A whole number, including zero and the negative numbers. The fixed gap and the powers in the theorem are integer quantities.
Exponential Diophantine equation: A Diophantine equation in which unknowns occur as exponents. The equation between two powers separated by a fixed difference is an exponential Diophantine equation.
Alan Baker: British mathematician who developed effective methods for bounding linear forms in logarithms. His estimates supplied the central tool in Tijdeman's proof.
S-unit equation: An equation involving sums of elements whose prime factors lie in a fixed finite set. Related finiteness methods connect exponential Diophantine problems to this equation class.
Exponentiation: The operation of raising a number to a specified power. Perfect powers are generated by exponentiation with integer bases and exponents.
Perfect power: An integer equal to an integer base raised to an integer exponent greater than one. The theorem concerns pairs of these integers with a prescribed gap.
Transcendence theory: The study of whether numbers are algebraic or transcendental and of quantitative relations among them. The theorem imports transcendence methods into a problem about integer powers.