Showing what's behind the curtain

Finding the maximum remainder when (a ā 1)^{n} + (a + 1)^{n} is divided by a^{2}.

Investigate the game of chance involving colored discs.

Find the maximum remainder when (`p` ā 1)^{n} + (`p` + 1)^{n} is divided by `p`^{2} for prime `p`.

Find the kth element in a sorted list of radicals.

Finding square sums that are palindromic.

Find the 2000th hexagonal tile in which the differences between it and its 6 neighbors yield 3 primes.

Investigating minimal repunits that divide by n.

Finding composite values, n, for which nā1 is divisible by the length of the smallest repunits that divide it.

Determining the first forty prime factors of a very large repunit.

Investigating which primes will never divide a repunit containing 10**n digits.