The Prime Number Formulas

There are many proposed partial prime number formulas, however, no formula can generate all prime numbers. Here we show three formulas which can obtain the entire prime numbers set from the positive integers, based on the Möbius function plus the “omega” function, or the Omega function, or the divisor function. The history of searching for a prime number formula goes back to the ancient Egyptians. There have been many proposed partial prime number formulas (e.g., Euler: P(n) = n2 +n+41), however, no formula can generate all prime numbers. Here we show Pk(n) ≡ n · a$(n) to pick up the prime numbers from the positive integers, which is based on the prime number definition of P ≡ P · 1 (i.e., a prime number can only be divided by 1 and itself). Let P, b ∈ Z, for a natural number n ≡ P · b, which is a prime number if b = 1 and a non-prime number if b 6= 1. The Möbius function The Möbius function is the sum of the primitive n-th roots of unity.[1]