proof of equivalence of Fermat’s Last Theorem to its analytic form

Consider the Taylor expansionMathworldPlanetmath of the cosine function. We have




For r>x,y the sequenceMathworldPlanetmath ar is decreasing as the denominator grows faster than the numerator. Hence for s>x,y the sequence As is increasing as As+4=As+as+2-as+4 and as+2>as+4. So if AN>0 for some N>x,y, we have 2-cosx-cosy>0. Conversely if no such N exists then As0 for s>x,y, so its limit, 2-cosx-cosy, is also less than or equal to 0. However as this expression cannot be negative we would have 2-cosx-cosy=0.

Similarly for r>z the sequence br is decreasing, and for s>z the sequence Bs is increasing. So if BM>0 for some M>z we have 1-cosz>0. Conversely if no such M exists then 1-cosz0. However as this expression cannot be negative we would have 1-cosz=0.

Note that 2-cosx-cosy=0 precisely when x,y2π. Also 1-cosz=0 precisely when z2π.

So the form of the theorem may be read:

If for positive reals x,y,z we have xn+yn=zn for some odd integer n>2, then either x or y not in 2π or z not in 2π.

Clearly this only fails if for positive integers a,b,c and some odd n>2, we have


Dividing through by (2π)n we see that an+bn=cn.

Conversely suppose we have non-zero integers satisfying an+bn=cn for some n>2. If n=4k we have (ak)4+(bk)4=(ck)4, contradicting example of Fermat’s last theorem. Hence if n is even we may replace a,b,c with a2,b2,c2 and n with n/2, which will be odd and greater than 1 (and hence greater than 2 as it is odd). So without loss of generality we may assume n odd.

Finally replace a,b,c with their absolute valuesMathworldPlanetmathPlanetmathPlanetmathPlanetmath and if reorder to obtain a positive integer solution. This would be a counterexample to the form of the theorem as stated above.

Title proof of equivalence of Fermat’s Last Theorem to its analyticPlanetmathPlanetmath form
Canonical name ProofOfEquivalenceOfFermatsLastTheoremToItsAnalyticForm
Date of creation 2013-03-22 16:19:04
Last modified on 2013-03-22 16:19:04
Owner whm22 (2009)
Last modified by whm22 (2009)
Numerical id 7
Author whm22 (2009)
Entry type Proof
Classification msc 11D41