Problem of the Week #20: Monday May 22nd, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
A perfect number is a natural number that is equal to the sum of all of its proper divisors (divisors that are less than the number itself). For example, 6 is the smallest perfect number because its proper divisors, 1, 2, and 3, add up to exactly 6.
a). The second smallest perfect number is between 20 and 30. Find this number.
The values of subsequent perfect numbers grow surprisingly quickly. There are now only 51 perfect numbers that have been found, the largest of which has close to 50 million digits. All 51 of the currently known perfect numbers are even, and humans have yet to prove that there are no odd perfect numbers. However, we can start excluding numbers with more specific properties:
b). Show that it’s impossible for a perfect number to be the square of an odd number.
c). Is it possible for a perfect number to be the square of an even number?
Given a natural number n, if the sum of all of n’s proper divisors is greater than n (instead of equal to, as in the case of n being a perfect number), then n is called an abundant number. If the sum is less than n, then n is called a deficient number.
d). Show that all prime numbers are deficient numbers.
e). Find the smallest abundant number.
f). Look up terms related to perfect, abundant, and deficient numbers on the internet.
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.