**Known as the 'Phi Function', tells how many numbers less than a number 'N' have no common factors (other than 1) with 'N'** There's a formula for calculating this, but it was complicated. More important is what it is. $\phi(8) = 4$ Phi of 8 equals 4 _because_: 8's factors are `[1, 2, 4, 8]`, and of the numbers less than 8: - 1: factors: `[1]` ✅ - 2: factors `[1, 2]` ❌ - shares `2` - 3: factors `[1, 3]` ✅ - 4: factors `[1, 4]` ❌ - shares `4` - 5: factors `[1, 5]` ✅ - 6: factors `[1, 2, 3, 6]` ❌ - shares `2` - 7: factors `[1, 7]` ✅ And there are **4** numbers that share no common factors other than 1. Another way of saying that is that there are 4 numbers less than 8 whose [[Greatest Common Factor|GCF]] is '1'. Phi of any of the [[Prime Numbers]] is just that number - 1. Phi of 7 is 6. Phi of 11 is 10. Weirdly, Phi is _multiplicative_, which means: $\phi(A*B)=\phi(A)*\phi(B)$ THIS is (I think) what makes it useful in [[Public Key Encryption]]. **** # More ## Source - [Kahn Academy](https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/euler-s-totient-function-phi-function) ## Related - [[Public Key Encryption]] - [[Primitive Root]]