Show that all sums of two consecutive odd prime
numbers have at least three prime factors, not necessarily distinct.
Example: 3+5=8. 8's factors: 2,2,2.
The question involves a little bit of attempted misdirection, as this can be proven for any pair of consecutive odd numbers, not just primes.
If the even number that separates them is 2n, then their sum is 4n, so the sum can be factored as 2*2*n, and n must have at least 1 prime factor. The 2 two's get us up to at least 3 prime factors.
Edited on February 5, 2015, 2:29 pm