406 H. F. TALBOT ON FERMAT’S THEOREM. 
Extension of FERMAT’S Theorem. 
Always supposing that a is a prime number, and that b—c is greater than 
unity, the theorem a"=b"—c" (impossible), may be extended to a much more ge- 
neral theorem, viz., that a"=b"—c" is impossible, provided that m is less than n. 
Demonstration. Let a"=k", where we no longer suppose / to be an integer. 
Therefore since m <n, a must be greater than /. But k"=b"—c" by hypothesis; 
therefore b"=c"+k", which is less than c+4|"; therefore 6 is less than c+, and 
b—cislessthan k. A fortiori, b—c is less than a. 
But in the given equation a=b"—c", since @ is prime, and b—c divides b’—c’, 
therefore 5—c=a, or else is divisible by a, a number which we have shown to be 
greater than itself, which is impossible. Therefore a=b"—c" is impossible un- 
less m >n. But if m >n, it is possible. 
Example. 3°=6? —3°, where a is prime and b—c greater than 1, but m > n. 
By an analogous method we obtain the extended theorem No. II. 
If 7 is an odd number, a"=b"+c" is impossible, provided that a is prime, and 
MK. 
We have hitherto supposed @ to be prime, whereas Ferma’s theorem has no 
such limitation; it remains, therefore, to enquire how far the present extended 
theorems are true when a is not a prime number. 
In conclusion, we may oberve that the ancients themselves had discovered the 
possibility of the equation a?=0? +c’. 
But from what precedes, we may deduce the following theorems concerning it. 
1. If @=2+?, and ¢ is a prime number, then a—d is always =1. 
2. If a2 =b? +, b and ¢ cannot both be prime numbers. For because cis prime, 
it follows that b=a—1. 
And because 6 is prime, therefore c=a—1, therefore b=c, and a?=2 3B’, 
But this is impossible, since one square cannot be double of another, in integer 
numbers. 
Examples. 5°=4? +8, and 3 being prime, we have 5—4=1. 
Again, 13?=12? + 5?, and 5 being prime, we have 13—12=1. 
Again, 25°=24°+7°, and 7 being prime, we have 25—24=1. 
The converse, however, is not true. For if a?=0?+¢?, and a—b=1, it by no 
means follows that either } or c isa prime. For example, 221?=220?+212, none 
of which numbers are primes. 
