What’s the point to searching for the largest prime number?
In: 12
By the way, there is no largest prime number. There is *always* a bigger prime.
The proof for this uses the technique of “proof by contradiction”, where we imagine that we found a largest prime and then show that this is impossible:
1. Imagine that the largest prime is the number **n**, and the full set of prime numbers is **{2, 3, 5, 7, …, n}**. This set is large, but finite. Call this set **P**.
2. Multiply together all the numbers in **P**. We can call the result **ΠP**. (The Π stands for “product”.)
3. Then add one, giving **ΠP+1**.
4. The number **ΠP+1** is not divisible by any of the numbers in **P**, which means that **ΠP+1** is prime.
5. However, **ΠP+1** is not in **P**. Therefore, **P** is not really the full set of primes.
6. **ΠP+1** is also greater than **n**. Therefore, **n** is not really the largest prime.
(Step 4 is a little tricky. We know **ΠP+1** is not divisible by *any* of the numbers in **P** because **ΠP** is divisible by *all* of those numbers. For instance, we know **ΠP+1** is not divisible by 2 because it’s one more than a multiple of 2; same for 3, 5, and every other number in **P**.)
Latest Answers