Prime Numbers – True/False1. There is a formula for the prime. 2. There is a formula for the number of primes below . 3. There are infinitely many primes. 4. There are infinitely many twin primes. 5. All primes are one more or less than a multiple of 6. 6. Any integer greater than 2 can be written as the sum

of two primes. 7. Every integer greater than 1 can be written uniquely as

the product of primes. 8. $100,000 was offered to factorise a 309 digit number.

Prime Numbers – True/False

1. There is a formula for the prime.

FALSE-ishWe know some about what it isn’t… produces primes for to The rounded down part of is prime for all , but the catch is that we don’t know what is (and currently can only calculate it using primes, so it’s a bit of a circular formula)

Prime Numbers – True/False

2. There is a formula for the number of primes below .

True-ishMathematicians are working on it, and there are better and better ways of finding new primes, but so far there is no easy way to find the number of primes below . We do have a name for the formula: . And we know that for , so that’s a start…

Prime Numbers – True/False

3. There are infinitely many primes.

TrueWe 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.

UnknownTwin 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.

Prime Numbers – True/False

5. All primes are one more or less than a multiple of 6.

Nearly TrueAll integers can be written as or . Since and numbers all divide by 2, and divides by 3, all primes apart from and must be of the form or .

Prime Numbers – True/False

6. Any integer greater than 2 can be written as the sum of two primes.

UnknownKnown as Goldbach’s Conjecture, we suspect this to be true (no counter-examples have been found), but it is still unproven.

Prime Numbers – True/False

7. Every integer greater than 1 can be written uniquely as the product of primes.

TrueThis fact is so important it has its own name:The Fundamental Theorem of Arithmetic (FTA)The proof follows from the basic properties of division, and follows from the idea that a factor of is a factor of either or .

Prime Numbers – True/False

8. $100,000 was offered to factorise a 309 digit number.

TrueRSA 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.