One of the most beautiful unsolved problems in mathematics, connecting the "size" of a set to the patterns hiding within it.
"If the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic progressions."
An arithmetic progression (AP) is simply a sequence of numbers where the difference between consecutive terms is constant. You've seen these your whole life!
Each number is 2 more than the previous.
Each number is 7 more than the previous.
The length of an arithmetic progression is simply how many terms it has. The AP "3, 5, 7" has length 3. The AP "2, 5, 8, 11, 14" has length 5.
When we say a set contains "arbitrarily long" arithmetic progressions, we mean: no matter what length you pick (100? 1,000,000?), you can find an AP of that length somewhere in the set.
This is where it gets interesting. The reciprocal of a number n is simply 1/n.
When you add up infinitely many numbers, one of two things happens:
If the sum of reciprocals diverges, it means the set is "dense enough" in some mathematical sense. The conjecture says: if a set is dense enough (divergent reciprocals), it must contain patterns (arbitrarily long APs).
Click numbers to add them to your set. Watch the sum of reciprocals and discover arithmetic progressions!
The prime numbers are a perfect example! The sum of reciprocals of primes diverges (proved by Euler in 1737):
So the conjecture predicts: primes should contain arbitrarily long arithmetic progressions.
And indeed, in 2004, Ben Green and Terence Tao proved exactly this! Their celebrated Green-Tao Theorem showed that for any length k, there exists an arithmetic progression of k primes.
"The primes contain arbitrarily long arithmetic progressions."
While Green-Tao proved it for primes specifically, the general conjecture — that ANY set with divergent reciprocal sum contains arbitrarily long APs — remains unproven. That's what the $5,000 prize is for!
Sets with positive density contain arbitrarily long APs. Won the Abel Prize!
Primes contain arbitrarily long APs. A landmark result!
Various special cases proven. Connections to ergodic theory established.
The full statement for arbitrary sets with divergent reciprocal sums. Still open!
The difficulty lies in the gap between "positive density" and "divergent reciprocal sum." A set can have divergent reciprocals while being extremely sparse — the primes are an example. We need new techniques to bridge this gap.
This conjecture sits at the heart of additive combinatorics, a field that studies the additive structure hidden in sets of numbers.
Solving this problem would deepen our understanding of how structure emerges from density — a fundamental question across mathematics.
"Problems are the lifeblood of mathematics."