Fermat number
and "F""j"; hence "a" divides their difference, 2. Since "a" &gt; 1, this forces "a" = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each "F""n", choose a prime factor "p""n"; then the sequence {"p""n"} is an infinite sequence of distinct primes.