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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Click on the name of each mathematical conjecture. Except where noted, these are open problems and mathematicians do not know if they are true or false. Some famous former conjectures are included.

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