What are some mind-blowing facts about mathematics?

Prime Generating Polynomials :

The polynomial, n2+n+41$n^2+n+41$ can be used to produce 40 primes for consecutive integer values 0≤n≤39. This property was discovered by Euler. Similarly, the incredible formula, n2−79n+1601$n^2-79n+1601$ was discovered, which produces 80 primes for the consecutive values 0≤n≤79!

Kaprekar’s Constant :

6174 is known as Kaprekar’s Constant, after the Indian Mathematician D.R. Kaprekar.

1. Take any four-digit number, using at least two different digits (Leading zeros are allowed.)
2. Arrange the digits in descending and then in ascending order, to get two four-digit numbers, adding leading zeroes if necessary.
3. Subtract the smaller number from the larger number and go back to step 2.

The above process will always reach the fixed number, 6174, taking at most 7 iterations. Try it yourself! (Note that after the first subtraction or the subsequent subtractions, the result obtained is always a multiple of 9!)

Collatz Conjecture :

Take any positive integer n. If n is even, divide it by 2 to get n2${\displaystyle \frac{n}{2}}$. If n is odd, multiply it by 3 and add 1 to get 3n+1$3n+1$. Repeat the process endlessly and no matter what number you start with, you will eventually reach 1.

Whether or not it is possible for all positive integer initial values to eventually reach 1, is one of the well known unsolved problems in mathematics. Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered $500 for its solution.

Fermat’s Last Theorem (Fermat’s Conjecture) :

It states that no three positive integers a, b, and c satisfy the equation an+bn=cn$a^n+b^n=c^n$ for any integer value of n greater than two.

The first successful proof was released in 1994 by British Mathematician Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians after the conjecture was proposed. The proof is over 150 pages long and took almost 7 years to develop. Prior to Wiles's proof, thousands of incorrect proofs were submitted, amounting to roughly 10 feet (3 meters) of correspondence.

It is among the most notable theorems in the history of mathematics and prior to its proof, it was in the Guinness Book of World Records as the "most difficult mathematical problem", one of the reasons being that it has the largest number of incorrect and unsuccessful proofs!