The proof of Fermat's Last Theorem

Abstract

(i) Instead of xⁿ+yⁿ=zⁿ, we use

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadI% hacqGHsislcaWGIbGaaiykamaaCaaaleqabaGaamOBaaaakiabgUca% RiaadIhadaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaGGOaGaamiEai% abgUcaRiaadggacaGGPaWaaWbaaSqabeaacaWGUbaaaaaa!448D!\[(x - b)^n + x^n = (x + a)^n \]

as the general equation of Fermat's Last Theorem (FLT), where a and b are two arbitrary natural numbers. By means of binomial expansion, (0.1) can be written as

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa% aaleqabaGaamOBaaaakiabgkHiTmaaqahabaGaaiikamaaDeaaleaa% caWGYbaabaGaamOBaaaakiaacMcacaWG4bWaaWbaaSqabeaacaWGUb% GaeyOeI0IaamOCaaaakiaacUfacaWGHbWaaWbaaSqabeaacaWGYbaa% aOGaeyOeI0IaaiikaiabgkHiTiaadkgacaGGPaWaaWbaaSqabeaaca% WGYbaaaOGaaiyxaiabg2da9iaaicdaaSqaaiaadkhacqGH9aqpcaaI% XaaabaGaamOBaaqdcqGHris5aaaa!514F!\[x^n - \sum\limits_{r = 1}^n {({}_r^n )x^{n - r} [a^r - ( - b)^r ] = 0} \]

Because a^r -(-b)^r always contains a+b as its factor, (0.2) can be written as

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa% aaleqabaGaamOBaaaakiabgkHiTiaacIcacaWGHbGaey4kaSIaamOy% aiaacMcadaaeWbqaaiaacIcadaqhbaWcbaGaamOCaaqaaiaad6gaaa% GccaGGPaGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaadkhaaaGc% cqaHgpGzdaWgaaWcbaGaamOCaaqabaGccqGH9aqpcaaIWaaaleaaca% WGYbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa!4F21!\[x^n - (a + b)\sum\limits_{r = 1}^n {({}_r^n )x^{n - r} \phi _r = 0} \]

where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaadkhaaeqaaOGaeyypa0ZaaSGbaeaacaGGBbGaamyyamaa% CaaaleqabaGaamOCaaaakiabgkHiTiaacIcacqGHsislcaWGIbGaai% ykamaaCaaaleqabaGaamOCaaaakiaac2faaeaacaGGOaGaamyyaiab% gUcaRiaadkgacaGGPaaaaaaa!4712!\[\phi _r = {{[a^r - ( - b)^r ]} \mathord{\left/ {\vphantom {{[a^r - ( - b)^r ]} {(a + b)}}} \right. \kern-\nulldelimiterspace} {(a + b)}}\] are integers for r=1, 2, 3, ..., n (ii) Lets be a factor of a+b and let (a+b)=sc. We can use x=sy to transform (0.3) to the following (0.4)

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaado% hacaWG5bGaaiykamaaCaaaleqabaGaamOBaaaakiabgkHiTiaadoha% caWGJbWaamWabeaadaaeWbqaaiaacIcadaqhbaWcbaGaamOCaaqaai% aad6gaaaGccaGGPaGaaiikaiaadohacaWG5bGaaiykamaaCaaaleqa% baGaamOBaiabgkHiTiaadkhaaaGccqaHgpGzdaWgaaWcbaGaamOCaa% qabaGccqGHRaWkcaWGUbGaam4CaiaadMhacqaHgpGzdaWgaaWcbaGa% amOBaiabgkHiTiaaigdaaeqaaaqaaiaadkhacqGH9aqpcaaIXaaaba% GaamOBaiabgkHiTiaaikdaa0GaeyyeIuoaaOGaay5waiaaw2faaiab% g2da9iaadohacaWGJbGaeqOXdy2aaSbaaSqaaiaad6gaaeqaaaaa!6181!\[(sy)^n - sc\left[ {\sum\limits_{r = 1}^{n - 2} {({}_r^n )(sy)^{n - r} \phi _r + nsy\phi _{n - 1} } } \right] = sc\phi _n \]

(iii) Dividing (0.4) by s² we have

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaCa% aaleqabaGaamOBaiabgkHiTiaaikdaaaGccaWG5bWaaWbaaSqabeaa% caWGUbaaaOGaeyOeI0Iaam4yamaadmqabaWaaabCaeaacaGGOaWaa0% raaSqaaiaadkhaaeaacaWGUbaaaOGaaiykaiaadohadaahaaWcbeqa% aiaad6gacqGHsislcaWGYbGaeyOeI0IaaGymaaaakiaadMhadaahaa% Wcbeqaaiaad6gacqGHsislcaWGYbaaaOGaeqOXdy2aaSbaaSqaaiaa% dkhaaeqaaOGaey4kaSIaamOBaiaadMhacqaHgpGzdaWgaaWcbaGaam% OBaiabgkHiTiaaigdaaeqaaaqaaiaadkhacqGH9aqpcaaIXaaabaGa% amOBaiabgkHiTiaaikdaa0GaeyyeIuoaaOGaay5waiaaw2faaiabg2% da9maalaaabaGaam4yaiabeA8aMnaaBaaaleaacaWGUbaabeaaaOqa% aiaadohaaaaaaa!6482!\[s^{n - 2} y^n - c\left[ {\sum\limits_{r = 1}^{n - 2} {({}_r^n )s^{n - r - 1} y^{n - r} \phi _r + ny\phi _{n - 1} } } \right] = \frac{{c\phi _n }}{s}\]

On the left side of (0.5) there is a polynomial of y with integer coefficients and on the right side there is a constant cφ/s. If cφ/s is not an integer, then we cannot find an integer y to satisfy (0.5), and then FLT is true for this case. If % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca% WGJbGaeqOXdy2aaSbaaSqaaiaad6gaaeqaaaGcbaGaam4Caaaaaaa!3AC4!\[{{c\phi _n } \mathord{\left/ {\vphantom {{c\phi _n } s}} \right. \kern-\nulldelimiterspace} s}\] is an integer, we may change a and c such the % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca% WGJbGaeqOXdy2aaSbaaSqaaiaad6gaaeqaaaGcbaGaam4CaaaacqGH% GjsUaaa!3C8B!\[{{c\phi _n } \mathord{\left/ {\vphantom {{c\phi _n } s}} \right. \kern-\nulldelimiterspace} s} \ne \] an integer.