7,140
Largest number that's both triangular and tetrahedral.
945
Smallest odd abundant number.
3,240
Number of positions a Rubik's cube can be in after three moves.
3,003
Only number (othter than 1) known to appear eight times in Pascal's Triangle.
Hippasus was a Greek Mathematician who worked with Pythagoras. Hippasus might have discovered the irrationality of √2, and the other Pythagoreans might have drowned him for it (their philosophy did not recognize the existence of irrationals). There's very little historical evidence for this, but it's a neat story and is often retold despite its unclear accuracy. Vihart has an excellent video about both the math and the history involved.
For which integers k are there integer solutions to x3+y3+z3 = k? This was asked at least as early as 1850, but even today there are still some relatively small k for which the answer is unknown! Solutions for k = 74 were found in 2016, and for k = 33 and k = 42 in 2019 (with all three of x, y, z having at least 16 digits!). As of 2021, it is still unknown whether any solutions exist for k = 114.
In the 1500s, Niccolò Tartaglia determined how to solve cubics of the form x3 + mx = n. Scipione dal Ferro had previously solved this problem but told only his student Antonio Fior. Fior challenged Tartaglia to a public contest in which each tried to solve the most cubics quickly. Tartaglia won. Girolamo Cardano then convinced Tartaglia to teach him how to do this, promising not to break Tartaglia's secret. But Cardano broke his word and published the method in 1545, thus letting the world in on the secret to these mathematical puzzles.
For centuries, many mathematicians believed that a regular heptadecagon (17-gon) could not be drawn with only a straightedge and compass. But in 1796, Carl Gauss constructed one and, furthermore, proved exactly which n-gons are constructibe, which relates to primes of the form 2(2k)+1. Pierre de Fermat saw that 2(20)+1 = 3 through 2(24)+1 = 65,537 were all prime and claimed that all 2(2k)+1 were prime, but in 1732 Euler showed that 2(25)+1 is not prime. Today, it is still unknown if 2(2k)+1 is prime for any k > 4.