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Catalan’s Conjecture

The main mathematical news

2002: Preda Mihailescu proved Catalan’s conjecture – 8 and 9 are the only pair of consecutive perfect powers, thereby implying that no Wieferich prime pair actually solves the equation mp-nq=±1.

2000: Preda Mihailescu proved that any solutions p, q for mp-nq=±1 other than 2, 3 must be a Wieferich prime pair (=rare).

1976: For values of e245< p, q there are no solutions for the equation mp-nq=±1.

1960: To solve mp-nq=1, p must divide n and q must divide m.

1951: For any pair m, n there is at most one solution for the equation mp-nq=1.

1844: Catalan’s conjecture – 8 and 9 are the only pair of consecutive perfect powers.

1738: Euler proved that if the exponents are 2,3, the only consecutive powers are 23 and 32.

1320: Gersonides proved that if the bases are 2,3, the only consecutive powers are 23 and 32.

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Additional Theorems / conjectures / Open questions

·  A perfect square cannot be the successor/predecessor of a perfect power (except 8,9).

·  There cannot be three/four consecutive perfect powers.

·  A perfect power cannot be the successor/predecessor of a perfect cube (except 8,9).

·  Is it enough to prove the conjecture for primes? Why/why not?

·   Why continue searching for a proof if computers can cover the search space?

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The main mathematical concepts / Principles

Elementary Algebra MSC2010#97H20

–        Power, base, exponent

–        Prime number

–        Successor and predecessor

 Logic (MSC2010#97E30)

–        Proof by exhaustion

–        Proof by contradiction

–        Conjecture vs proof

Number theory (MSC2010#97F60)

–        Fermat’s Last Theorem

–        Catalan’s conjecture

–        Wieferich primes

–        Diophantine equations

–        Pythagorean triples

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