64 (number) 64 (number) 64 (sixty-four) is the natural number following 63 and preceding 65. Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number that is both a perfect square and a perfect cube. Since
Proof that e is irrational and "b" such that "e" = ⁄. Define the number To see that if "e" is rational, then "x" is an integer, substitute "e" = ⁄ into this definition to obtain The first term is an integer, and every fraction in the sum is actually an integer because "n" ≤ "b" for each term. Therefore, "x" is an integer. We now prove that . First, to prove that "x" is strictly positive, we insert the above series representation of "e" into the definition of "x" and obtain because all the terms are strictly positive. We now prove that "x" <