The interesting number paradox is a simple way to reinforce that sometimes the best way to prove something is to actually prove its contrary.
My curiosity about interesting numbers stemmed from a variety of online chatter about how the year 2025 was mathematically interesting:
- 2025 is a perfect square (45 x 45 = 2025).
- 2025 is the product of prime factors (3 4 x 5 2).
- 2025 is the sum of the cubes of the first nine natural numbers (1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2025).
- … and several more.
The concept of interestingness is not actually defined mathematically but is generally agreed to denote something unique or noteworthy about a number. For example:
- 1 is the multiplicative identity (any number multiplied by 1 equals that number)
- 2 is the only even prime number
- 3 is the number of dimensions humans can see
- 4 is the smallest composite number (divisible by numbers other than 1 and itself)
- … and so on.
This interactive site describes itself as the almanac of interesting numbers.
The interesting number paradox starts with a simple question: What is the smallest number that is not interesting? There are lots of suggestions for the number (here, here, here, and here as representative examples).
However, whatever you decide is the smallest uninteresting number, that fact automatically makes the number interesting so it’s no longer uninteresting. As a result, there cannot be a smallest uninteresting number because that would make it interesting.
Like the Raven Paradox, sometimes we cannot find the information to backup a claim we are trying to make. Instead, we prove the contrary cannot exist.
That is the interesting number paradox.
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