  We present closed forms for several functions that are fundamental in number
theory and we explain the method used to obtain them. Concretely, we find
formulas for the p-adic valuation, the number-of-divisors function, the
sum-of-divisors function, Euler's totient function, the modular inverse, the
integer part of the root, the integer part of the logarithm, the multiplicative
order and the discrete logarithm. Although these are very complicated, they
only involve elementary operations, and to our knowledge no other closed form
of this kind is known for the aforementioned functions.
