It has been proven that there is no largest Prime number. You can find a list of the largest KNOWN prime numbers here: http://en.wikipedia.org/wiki/Largest_known_prime_number
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No "largest" composite number exists.
On Jan. 25, 2013, the largest known prime number, 257,885,161-1, was discovered on Great Internet Mersenne Prime Search (GIMPS) volunteer Curtis Cooper's computer. The new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits.
On Jan. 25, 2013, the largest known prime number, 2^57,885,161-1, was discovered on Great Internet Mersenne Prime Search (GIMPS) volunteer Curtis Cooper's computer.
I assume you mean twin prime. It is currently unknown whether or not there are an infinite number of twin primes in which case there would be no largest twin prime. The largest discovered is 65516468355 · 2333333 ± 1 (so the bigger of the two is 65516468355 · 2333333 + 1).
indian mathematician aryabhatta