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Prime Numbers β True/False There is a formula for the ππ‘β prime. There is a formula for the number of primes below π. There are infinitely many primes. There are infinitely many twin primes. All primes are one more or less than a multiple of 6. Any even 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. 1. 2. 3. 4. 5. 6. Prime Numbers β True/False 1. There is a formula for the ππ‘β prime. FALSE-ish We know some about what it isnβtβ¦ π2 β π + 41 produces primes for π = 0 to π = 40 The rounded down part of π΄3π is prime for all π, but the catch is that we donβt know what π΄ is (and currently can only calculate it using primes, so itβs a bit of a circular formula) Prime Numbers β True/False 2. There is a formula for the number of primes below π. True-ish Mathematicians 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 π. We do have a name for the formula: π(π). And we know that π π < π π < 1.25506 for π > 10, so thatβs a startβ¦ ln π ln π Prime Numbers β True/False 3. There are infinitely many primes. True We can prove this by assuming there arenβt: Multiply all the primes together, then add 1. This number doesnβt divide by any of the primes, but all numbers greater than 1 either are prime or divide by primes. So this number either divides by primes not in our list, or is itself a prime not in our list. Contradiction! Prime Numbers β 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. Prime Numbers β True/False 5. All primes are one more or less than a multiple of 6. Nearly True All integers can be written as 6π, 6π + 1, 6π + 2, 6π + 3, 6π + 4 or 6π + 5. Since 6π, 6π + 2 and 6π + 4 numbers all divide by 2, and 6π + 3 divides by 3, all primes apart from 2 and 3 must be of the form 6π + 1 or 6π + 5. Prime Numbers β True/False 6. Any even integer greater than 2 can be written as the sum of two primes. Unknown Known as Goldbachβs Conjecture, we suspect this to be true (no counter-examples have been found), but it is still unproven. Prime Numbers β True/False 7. Every integer greater than 1 can be written uniquely as the product of primes. True This 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 ππ is a factor of either π or π. Prime Numbers β True/False 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 βPublic Keyβ, 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.