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单词 ProofOfEquivalenceOfFermatsLastTheoremToItsAnalyticForm
释义

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


Consider the Taylor expansionMathworldPlanetmath of the cosine function. We have

lims(As)=2-cosx-cosy

and

lims(Bs)=1-cosz.

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 andas+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 thisexpression 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 isincreasing. So if BM>0 for some M>z we have 1-cosz>0.Conversely if no such M exists then 1-cosz0. However as thisexpression cannot be negative we would have 1-cosz=0.

Note that 2-cosx-cosy=0 precisely when x,y2π. Also1-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 ory 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

(2πa)n+(2πb)n=(2πc)n.

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. Ifn=4k we have (ak)4+(bk)4=(ck)4, contradictingexample of Fermat’s last theorem. Hence if n is even we may replace a,b,c witha2,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 mayassume n odd.

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

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更新时间:2025/5/4 17:03:29