3. There are infinitely many primes. True We can prove this by assuming there aren’t: Multiply all the primes together, then add 1. This number is one more than a multiple of any of them, so it doesn’t divide by any of the primes. Since any number either divides by other primes or is itself prime, this must be a new prime. But we used all of them: Contradiction!
Prime Numbers – True/False 4. There are infinitely many twin primes. Unknown Twin primes are primes 2 apart. It is conjectured that this is true (and even that there are an infinite number of primes 4 apart, 6 apart, 8 apart, etc) but not yet proven.
6. Any integer greater than 2 can be written as the sum of two primes. Unknown Known as Goldbach’s Conjecture, we suspect this to be true (no counter-examples have been found), but it is still unproven.
8. $100,000 was offered to factorise a 309 digit number. True RSA encryption is based on the difficulty of factorising large numbers. Two large primes are multiplied together to produce a ‘Public Key’, meaning anyone can encrypt data to send to you, but only you - the person who generated the public key - can decrypt it. The only known way to break the code is to factor the public key.