Mr, Dracli on Fermat's Undemonstrated Theorem. 287 

 similarly, the equation in Y givesiw lav/oq eooiiaia^ni imii "io 



1 + SAil-g-) Jfij ^niJiimbA .88 



If w = 1, p = 1 =: gr; if w > 1, Af > 1, and as a? is positive, 

 Jt^z^ q^y'i therefore 2?'* is multiplied by a series of positive 

 terms =2Ai x a quantity less than unity; whilst the factor of 

 ^ is similar, but = S Aj x quantity > 1, .\p^ > g'"* "^ order 

 that the respective products shall =1. 

 .Combining the last found equations, vk'e obtain by the b|^ 



P \^z (~~^ ^— ~E H)— \ jt j«ij_ »^ ~3— f+ 



the difference of these quantities is therefore zero. But put- 

 ting the coefficient of the eth term in the binomial develop- 

 ment = Bj, so that 



Pi — /J_ ^~^ 1— 2n 1+4;? — 2?n 1 +3n — 2in _^ \ 

 ^'~\n * 2w ' S» * 2m— 3w * 2zw — 2» Vi -V 



-. -: •■'KMr. v'd ,2'W 



l-2 z — 2.W 1— 2 2- l.w 



^ - ^ 2?w-« 277^^ ' "VS^ avoiViia 'i^AV oT 



the difference is expressible by t.^^rMa^TKaO 



when w=l,jp=g', w— 1=0, the equation is satisfied ; when 

 n exceeds unity, we have 1 — 2 « w = a negative quantity, and 

 consequently the factor within ( ) is positive, as is also 

 (^ -J-^)'"; but then Bj = Bj-i x the product of two negative 

 fractions, and is consequently of the same sign as B,_i or 

 Bj_2 ... or Bj, but Bj is negative, .'.0 = a series of exclusively 

 negative quantities^ which is impossible, unless a factor com- 

 mon to all = 0, which is here 1 — w, as the case of y = is of 

 course excepted. Consequently w= 1 is the only odd possible 

 value of this exponent. 



Case 2, w = 2 ?«, or x^'^ t= z^^ — y«. 



As the sum of two squares is never of the form 4 ? + 2 or 

 4 2 + 3, it follows that x must be even and y odd ; also is is 

 odd. Moreover, m must be odd, as we know that the sum of 

 two biquadrates cannot be a square, much less a biquadrate 



