The sum of the squares of the reciprocals of the positive integers is π2/6.
The sum of their fourth powers is π4/90.
The sum of their sixth powers is π6/945.
The area of the region under the Gaussian curve y = e–x2 is the square root of π.
The probability that two integers chosen at random will have no prime factor in common is 6/π2.
The integer 8 can be written as the sum of two squares of integers, m2 + n2, in four ways, when (m, n) is (2, 2), (2, -2), (-2, 2), or (-2, -2). The integer 7 can’t be written at all as the sum of such squares. Over a very large collection of integers from 1 to n, the average number of ways an integer can be written as the sum of two squares approaches π. Why?