Perhaps I shouldn't have said "large" or "notable". The point is that when a prime number is discovered meeting certain criteria it is included in public databases. This makes the case of an "illegal" such prime interesting.
No. Even then they're not notable. For example there are 10^97.6 primes with 100 digits, of which not all have been found (not enough storage space in the universe), but it's trivially easy to generate one at random.
I'm arguing that these numbers aren't notable just because they are large primes. They certainly aren't included in public databases just because they are large primes.
The example site you gave bigprimes.net has one main database, that includes the first 1.4 billion primes. These are all much smaller than the illegal primes mentioned on the wikipedia page (they have at most 11 digits, whereas the ones mentioned on wikipedia have 1000s of digits). It also has a list of the Mersenne primes, which are the largest known primes. The largest has 23249425 digits.
So primes with 11 digits or fewer are notable (because they are small), and primes with 23249425 digits or more are notable (because they are large), but primes with around 2000 digits are not notable.
Ok, thanks for explaining that. It sounds like the condition for notability in this case is the combination of size and algorithm used?
> Carmody created a 1905-digit prime, of the form k·256211 + 99, that was the tenth largest prime found using ECPP, a remarkable achievement by itself and worthy of being published on the lists of the highest prime numbers.
Oh, I didn't realize the page said that. I should have read it more carefully. You're right that it's a combination of size and method used. The Mersenne numbers are especially easy to prove prime, so it's also interesting to keep track of the largest primes proved by other methods. I think these days a 2000 digit prime wouldn't be notable in any case, but it sounds like it was back when Carmody was doing this, so this was in fact a good example of an illegal number that was independently interesting.
