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 π is irrational recursive integration by parts allows us to deduce that, if "a" and "b" are natural number such that = "a"/"b" and "f" is the polynomial function from [0,] into R defined by then: This last integral is 0, since "f" is the null function (because "f" is a polynomial function of degree 2"n"). Since each function "f" (with ) takes integer values on 0 and on and since the same thing happens with the sine and the cosine functions, this proves that "A"("b") is an integer. Since it is also greater than 0, it must be a natural number. But