First Record found of Prime Number was 300BC old briefing about their infinitude.
Their is 50000$ Prize for whoever Discover Largest Unknown Prime Number by GIMPS and EFF.Ofcourse with Computer Softwares, Various Test and Algorithm
All largest prime numbers discovered are mersenne prime which are "n" power of 2 less by 1 where "n" is also a prime.
There are Prime Numbers Categarize as Ramanujan Primes (Rn) which discussed in movie "The Man who knew Infinity".
The Largest Known Prime Number has 2crore 48lac 62thousands 42digits in it.