404 H. F. TALBOT ON FERMAT'S THEOREM. 



The case which admits of this simple demonstration, is that in which one of 

 the three numbers a, b, c, is a prime number ; and it divides itself into the two 

 following theorems : — 



Let a be any prime number, then, 



Theorem I. If n is any odd number greater than 1, the equation a n =b n + c is 

 impossible. 



TheoremM. If n is any number, odd or even, greater than 1, the equation 

 a n =b n — c n is impossible. 



But Theorem II. admits a case of exception, viz., that whenever b— c=l, the 

 theorem remains undemonstrated. When n=2, this case of exception actually 

 occurs, because a 2 = b 2 —c 2 is possible, although a be a prime number. For ex- 

 ample, when a =3, 3 2 = 5 2 -4 2 . 



Such a case of exception, however, does not occur when n=3, or n—i, or, 

 n=5, as Legendre has demonstrated. But that is no reason why it should not 

 occur with other values of n. And therefore it appears that the generality of 

 Fermat's theorem is assailable in this direction ; a fact which deserves the atten- 

 tion of mathematicians, especially as Fermat himself does not appear to have 

 adverted to it. 



In order to demonstrate these propositions, I will recall to mind some of the 

 leading principles of the Theory of Numbers. 



1. If a prime number does not divide either of the whole numbers A or B, it 

 does not divide their product AB. Legendre, p. 3, gives a very rigorous demon- 

 stration of this important theorem. 



2. When a number has been divided into its prime factors, it cannot be 

 divided into other prime factors different from the first ones. 



3. The product of any number of primes, cannot be equal to the product of 

 any number of other primes different from the first ones. 



And here it may be observed, that although these products cannot be equal, 

 nothing prevents them from approximating as closely as possible to equality, 

 i. e., differing by a single unit. For example, the product of the 



three primes 7 x 17 x 83 = 9877 



that of 2 x 11 x 449 = 9878 



that of 3 x 37 x 89 = 9879. 



These things being premised, we may proceed as follows : — 



Demonstration of Theorem I. 



Let us suppose, if possible, a"=b" + c". Then because n is an odd number, 



b n + c" is divisible by b + c. Let the quotient be Q. Therefore a" = b + c . Q. Now 



since a is prime, the first side of the equation is the product of the n factors 



ax ax ax &c. Consequently the second side of the equation is the product of 



