Value of phi for a prime power argument

If is prime and , then

Proof: Since is a prime number, the only possible values of are , and the only way to have is if is a multiple of , that is, , and there are such multiples not greater than . Therefore, the other numbers are all relatively prime to .

Proof of Euler's product formula

The fundamental theorem of arithmetic states that if there is a unique expression where are prime numbers

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