Some numbers follow a curious rule where, no matter what number you start with, repeating a calculation always lands you in the same spot. Stack numbers as dots and sometimes a triangle or square shape appears. This time, let's calculate Kaprekar's constant, figurate numbers, and Mersenne primes yourself and confirm these patterns.
Kaprekar's constant is a magical number that any four-digit number arrives at, no matter what you start with, if you repeat a fixed calculation. Figurate numbers are numbers that form shapes like triangles or squares when arranged as dots. Mersenne primes are the primes among numbers formed by multiplying 2 by itself several times and subtracting 1 — and they're deeply connected to perfect numbers.
Pick a four-digit number (as long as not every digit is the same), and subtract the number formed by arranging its digits smallest-to-largest from the one formed largest-to-smallest. Repeat this calculation, and it's guaranteed to reach 6174 within at most 7 steps.
Arrange numbers side by side as dots, and some numbers form a triangle, others a perfect square. Numbers like these are called figurate numbers. Pick a shape and a count (n) and check what number results.
A number formed by multiplying 2 by itself p times and subtracting 1 (2ᵖ−1) is called a Mersenne number. Even when p is prime, 2ᵖ−1 isn't always prime. But whenever 2ᵖ−1 is prime (a Mersenne prime), 2^(p−1)×(2ᵖ−1) is guaranteed to be a perfect number!