Catalan's conjecture

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The Catalan conjecture states that apart from the powers 2³ = 8 and 3² = 9 there are no other real powers that differ by exactly 1.

The conjecture was formulated in 1844 by the Belgian mathematician Eugène Charles Catalan. The calculation can be written as

3² − 2³ = 9 − 8 = 1

In 2002 the Romanian mathematician Preda Mihăilescu proved that the only solution of two consecutive powers in the natural numbers for

x^a − y^b = 1

with x, y, a, b > 1 actually is x = 3, a = 2, y = 2 and b = 3. That is why it is now also called the theorem of Mihăilescu.