These problems are for college undergrads after a first course in calculus. They are provided with solutions, and could be used by college professors as exercises or exam questions.
1. Digits of Pi/4
Prove that in base b, if b is an even integer, n > 3, and x = Pi/4, then the n-th digit of x, denoted as a(n), is given by the formula below. We start with n = 1 after the decimal point, for the first digit. Also show that the formula below is not valid if the base b is an odd integer, or if x is different from Pi/4.
where the brackets represent the integer part (also called floor) function.
Regardless of the number x in [0, 1] and the base b, the n-th digit a(n) of x can be computed as follows:
See here for details. Thus we have
Using the angle difference formula for sinus, the fact that n > 3, b is an even integer, and x = Pi/4, it simplifies to
The result for a(n) follows immediately.
2. Continued Fractions and Nested Square Roots
Let us consider the two following expressions, assuming a is a strictly positive real number:
Prove that x is an integer if and only if a is the product of two consecutive integers. Prove that the same is true for y.
Let's focus on the first case. The second case is almost identical. The strictly positive number x must satisfy x^2 = a + x, thus x = (1 + SQRT(1 + 4a)) / 2. In order for x to be an integer, 1 + 4a must be a perfect odd square, which is possible only if a is the product of two consecutive integers. For instance,
Note that the expansion of the number x = 2 in the nested square root numeration system, when x tends to 2, has all its "digits" equal to a = 1 * 2. See this spreadsheet for details. More on this here.