By René Schoof

ISBN-10: 1848001843

ISBN-13: 9781848001848

Eugène Charles Catalan made his well-known conjecture – that eight and nine are the single consecutive excellent powers of traditional numbers – in 1844 in a letter to the editor of Crelle’s mathematical magazine. 100 and fifty-eight years later, Preda Mihailescu proved it.

Catalan’s Conjecture provides this marvelous lead to a manner that's available to the complicated undergraduate. the 1st few sections of the booklet require little greater than a easy mathematical heritage and a few wisdom of straight forward quantity idea, whereas later sections contain Galois concept, algebraic quantity thought and a small volume of commutative algebra. the necessities, resembling the elemental evidence from the mathematics of cyclotomic fields, are all mentioned in the text.

The writer dissects either Mihailescu’s facts and the sooner paintings it made use of, taking nice care to pick streamlined and obvious types of the arguments and to maintain the textual content self-contained. merely within the evidence of Thaine’s theorem is a bit type box concept used; it really is was hoping that this software will inspire the reader to check the speculation further.

Beautifully transparent and concise, this publication will charm not just to experts in quantity thought yet to an individual drawn to seeing the appliance of the tips of algebraic quantity concept to a recognized mathematical problem.

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Consider the Fq [G]-module H = α ∈ Q(ζ p )∗ : ordr (α) ≡ 0 (mod q) for all primes r = p /Q(ζ p )∗ q . R. 1007/978-1-84800-185-5 7, C Springer-Verlag London Limited 2008 41 42 Catalan’s Conjecture In other words, an element α ∈ Q(ζ p )∗ (modulo qth powers) is contained in H if and only if the principal ideal generated by α is equal to aq pk for some k ∈ Z and some fractional ideal a. The Fq [G]-module H is the obstruction group that was mentioned in chapter 1. It plays an important role in our presentation of Mih˘ailescu’s proof.

Therefore, we have for certain integers u, v ∈ Z, y + 1 = u 2, y +1 = v2 . y+1 q The point (y, v) lies on the algebraic curve given by the equation Y q−1 − Y q−2 + . . − Y + 1 = V 2 . The algebraic curve described by this equation is irreducible. The associated homogeneous equation Y q−1 − Y q−2 Z + . . − Y Z q−1 + Z q−1 = V 2 Z q−3 describes a projective irreducible curve in P2 . This curve has a singularity at infinity. An equation that describes a curve with the same function field, but that is smooth in a neighborhood of infinity, is given by W 2 = 1 −U + .

This implies that 1−ζ pp is integral. Therefore, w is integral as well. x−ι(ζ ) Similarly, the fact that we have w q = (w/α)q = 1−ζ pp = −ι(wq ζ p ) implies that w is integral. Therefore, η is integral and hence is in Z[ζ p ]. 1. Therefore, w − w and η = (w − w )q are units as well. 8 Small p or q 49 (iii) Since η is a unit, the norm N (η) of η, taken from Q(ζ p ) to Q, is equal to a unit in Z. In other words, we have N (η) = ±1. We have the plus sign because the cyclotomic field Q(ζ p ) is complex.

### Catalan's conjecture by René Schoof

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