Catalan's conjecture
The catalansche guess is a set of the mathematical branch of number theory. It starts from the observation that one knows except the powers and no other real powers, which differ by exactly 1. Eugène Charles Catalan made in 1844 which was named after him catalansche presumption to the effect that there are no other real powers with this property:
Only after more than 150 years, this conjecture was proved in 2002 by Preda Mihăilescu.
History
Even before Catalan focused its attention on related issues. Approx. 1320 showed Ben Levi Gershon If powers of 2 and 3 differ by 1, then 8 and 9 are the only solutions.
Leonhard Euler (1707-1783) showed that there is for just the solution.
Catalans conjecture generalizes Euler's equation on general powers. His conjecture was published in 1844 in the "Journal for Pure and Applied Mathematics" as a letter to the editor.
Later they found some interesting partial results for the case that Catalans assertion is not true, ie that there are other non-trivial solutions of the equation.
So in 1976 showed Robert Tijdeman that at most finitely many numbers satisfy the equation.
1998 showed Ray Steiner following property for a possible solution: either and meet certain Teilbarkeitsbedingungen ( class number condition) or twice and are Wieferich primes, ie, they satisfy the condition
Maurice Mignotte arrived in 2000, an upper limit for solutions and to: q < 7.15 * 1011, p < 7.78 * 1016.
In April 2002, the then succeeded at the University of Paderborn finally employed Preda Mihăilescu the proof of catalanschen presumption with which this was given the status of a mathematical theorem.