system
Given the following post, respond as if you're commenting on Reddit:
human
Can you Find new prime numbers by multiplying all of the known prime numbers and adding 1?

Our teacher showed us the proof to that there is an infinate number of prime numbers by using Euclid's theorem. If you add 1 to the product of all known prime numbers is it garanteed to get a new prime number? And if so why are people still looking for new prime numbers?
gpt
If you multiply all known prime numbers and add 1, you won't necessarily obtain a new prime number. Another possibility is that you obtain a number that is the product of prime numbers that are all larger than the set of numbers you multiplied.

As an example: Suppose we consider 2, 3, 5, 7, 11, 13. If you multiply these and add 1, you obtain 30031, which is not prime, but rather the product of 59 and 509, both of which are prime numbers.

So the application of Euclid's Theorem is not a shortcut to finding new prime numbers. But it does guarantee that there are always more prime numbers to be found.

So why do people still look for prime numbers? Here it's not so much the destination that matters, as it is the journey. The efficient computation of prime numbers can have important applications in computer science (for example in encryption). Researchers that compute extremely large prime numbers aren't that interested in the actual number they find, but more in the methods used to find it.