A prime number is a number divisible only by 1 and itself: 2, 3, 5, 7, 11… They look like they follow no pattern, but look closely and surprising patterns turn up everywhere. The catch is that some of these patterns "always seem to hold whenever you check them," yet no mathematician, however famous, has managed to prove them. Some have stayed unsolved for centuries.
The first is twin primes — pairs of primes exactly 2 apart, like (3,5), (11,13), and (17,19). Primes get rarer and rarer as numbers grow larger, yet remarkably, twin primes just keep turning up. That naturally raises the question: "Are there infinitely many twin primes?" Most mathematicians believe the answer is yes, but no one has proven it yet. This is called the Twin Prime Conjecture.
The second is Goldbach's Conjecture, the claim that "every even number greater than 4 can be written as the sum of two primes" — for example, 28 = 5 + 23, or 100 = 3 + 97. Ever since Goldbach first shared this idea in a letter to Euler in 1742, computers have checked every even number up to over 4×10¹⁸ (4 quintillion) without finding a single exception. And yet, no one has proven that it holds "for every even number."
The third is Fermat primes. A Fermat number is 2 raised to the power of 2ⁿ, plus 1 (2^(2ⁿ)+1), and 17th-century mathematician Fermat predicted these numbers would always be prime. Indeed, for n=0,1,2,3,4 they came out to 3, 5, 17, 257, and 65537 — all prime. But in 1732, Euler found that the number for n=5, 4,294,967,297, divides evenly into 641 × 6,700,417, shattering Fermat's prediction. What's interesting is that this problem is unsolved in the exact opposite direction from the other two. For twin primes and Goldbach's conjecture, it's the claim that "there are infinitely many" that remains unproven; for Fermat primes, not a single prime has turned up among the thousands of candidates checked since n=5, and yet the claim that "there really is no 6th Fermat prime" also remains unproven.
What all three problems have in common is that no matter how many examples you check, that alone never proves something holds "in every case." Checking 4 quintillion even numbers doesn't guarantee that the very next even number will also work. In mathematics, a single counterexample is enough to bring an entire conjecture down, so no matter how overwhelming the evidence, a claim can't shed the label "conjecture" until it has a rigorous proof.
On our activity page, you can pick out twin prime pairs yourself, change an even number with a slider to see how many ways it splits into a Goldbach sum, and click through values of n to calculate for yourself whether each Fermat number is prime. Seeing it firsthand should give you a real feel for why mathematicians have been drawn to these problems for centuries.