Catalans Conjecture

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Catalans Conjecture

• Charlotte Turner
• Collingwood College

Catalans Conjecture
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Catalans Conjecture
What is the problem? Ingredients of the
proof Proving Catalans Conjecture
Catalans Conjecture
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The problem
1,4,8,9,16,25,27,32,36,49,64,81,100,
Are 8 and 9 the only consecutive powers?
Catalans Conjecture
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The problem
Catalans Conjecture
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Ingredients of the proof
Reduction to prime exponents V. A. Lebesgue
solved the case Ko Chao solved the case
Catalans Conjecture
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Ingredients of the proof
Theorems due to Mihailescu
Theorem I
Theorem II
Theorem III
Catalans Conjecture
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Proving Catalans Conjecture
p,q are odd primes
Use symmetry
Catalans Conjecture
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Proving Catalans Conjecture
Catalans Conjecture
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Proving Catalans Conjecture
Low exponents
Theorem IV
For odd primes p,q 41 there exist no non-zero
solutions x,y to Catalans equation
.
Catalans Conjecture
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Catalans Conjecture
Catalans Conjecture
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Mihailescus Theorem
Catalans Conjecture