3 ms·
You don't need two separate cases. Assume p1, ..., pn is a finite list of primes. The sum p1+...+pn+1 is divisible by a prime, because every natural number> 1
by Tainnor 5y ago
You don't need two separate cases.
Assume p1, ..., pn is a finite list of primes. The sum p1+...+pn+1 is divisible by a prime, because every natural number> 1 is. However, it's not divisible by p1,...,pn, hence there must be an additional prime not in the list.
(I think you're right though that GP's "contradiction" doesn't work)
- Tainnor 5y agotoo late to edit: p1 * ... * pn+1, of course. not plus.
- ivegotnoaccount 5y agoOh, you're right. Never thought of using "by definition, all numbers can be divided by a prime" tu merge the two cases. It's not that shorter, but is IMHO quite elegant, I'll remember it. Thanks for correcting me.
- Tainnor 5y agoWell, it's not by definition, but "every number is divisible by a prime" is fairly obvious (just keep dividing until you reach a prime) and can technically be proven by using (strong) induction.