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Another Conjecture Bites The Dust!

First the Four-Color Theorem, then Fermat's Last Problem, and then Kepler's Stacking Cannonballs Problem....all famous unsolved problems (and easily understood) that have been solved in my life-time. And there are others...I cannot keep up with all of this activity in mathematics!

In 2002, the Catalan Conjecture became Mihăilescu's Theorem. To understand it, consider all the whole numbers n (n>1) formed by a power greater than 1, e.g. 4, 8, 9, 16, 25, 27, 36, .... Note that 8 and 9 are sequential...are there any others? In formal terms, the Catalan Conjecture suggested that when considering the equation xp-yq=1, the only whole-number solution is 9-8 = 32-23=1.

Belgian mathematician Eugene Catalan (pictured) posed the conjecture in 1844. German mathematician Preda Mihăilescu finally proved the conjecture (i.e. it became a theorem) in 2002, using some very non-trivial mathematics.

So, another great conjecture has bitten the dust...and I missed it! But, it seems another related conjecture has risen from this dust...it is called Pillai's conjecture, but perhaps it too will be solved by the time you are reading this note. If so, you might want to explore the meaning of Catalan numbers, the Catalan solid, or the Catalan constant. Ah yes, mathematics is alive and well...and exciting!

Source: Adapted from Science News, May 25, 2002, pp. 324-325.