What’s the point to searching for the largest prime number?
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It’s been known since Euclid that there are infinitely many primes, so they aren’t searching for the largest one. And you should bear in mind that this is a pretty niche activity – it’s one of those things that gets disproportionate attention because it’s easy to explain and it involves impressively large numbers.
Finding individual large primes isn’t really that useful. Afaik, the very largest known primes are not actually used in cryptography at all. There are various things that are unknown about the primes, but it isn’t very likely that finding a few more will shed much light on those questions. But developing *techniques* to find large primes more efficiently is potentially useful and interesting, because those techniques might be applicable to other problems and might shed light on some theoretical issues (e.g. the development of the AKS primality test led to a proof that the problem of determining whether a number is prime is within a certain complexity class).
It’s been known since Euclid that there are infinitely many primes, so they aren’t searching for the largest one. And you should bear in mind that this is a pretty niche activity – it’s one of those things that gets disproportionate attention because it’s easy to explain and it involves impressively large numbers.
Finding individual large primes isn’t really that useful. Afaik, the very largest known primes are not actually used in cryptography at all. There are various things that are unknown about the primes, but it isn’t very likely that finding a few more will shed much light on those questions. But developing *techniques* to find large primes more efficiently is potentially useful and interesting, because those techniques might be applicable to other problems and might shed light on some theoretical issues (e.g. the development of the AKS primality test led to a proof that the problem of determining whether a number is prime is within a certain complexity class).
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**.)
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**.)
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**.)
There is no largest prime number just like there is no largest even or odd number or even just a largest number.
There is no largest prime number just like there is no largest even or odd number or even just a largest number.
There is no largest prime number just like there is no largest even or odd number or even just a largest number.
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