3² = 2³ + 1 is the only solution to aˣ = bʸ + 1 in integers greater than 1 (proved by Mihăilescu in 2002)

Let f(n) = n/2 for even n, and 3n + 1 for odd n. Starting from any positive integer, the orbit of f(n) will eventually reach 1

For every integer n > 1, there exist positive integers a, b, and c such that 1/a + 1/b + 1/c = 4/n

Every even integer greater than 2 equals the sum of two primes

Any square grid with an even number of spaces can be filled with +’s and −’s so all pairs of rows are exactly half-matching

There is a prime number between n² and (n+1)² for each positive integer n

There are infinitely many positive integers n for which 2ⁿ−1 is prime

Every convex shape in n dimensions can be covered by 2ⁿ smaller copies of itself

Each of n dots moving around a circle at different speeds will eventually be at least 1/(n+1) of the circle away from the rest

Every simply-connected closed 3-manifold is homeomorphic to the 3-sphere (proved by Perelman in 2003)

For every integer n > 2, at least half the positive integers less than n have an odd number of prime factors (disproved by Haselgrove in 1958)

There is a positive integer n such that only the number 1 appears more than n times in Pascal’s triangle

There are infinitely many prime numbers p for which 2p+1 is also prime

Every simple closed plane curve passes through all four corners of some square

There are infinitely many prime numbers p for which p+2 is also prime

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