Binomial coefficients coprime to 6

Imagine a vast, infinite grid of numbers, a hidden landscape where every point holds a secret about how things combine. This is the world of binomial coefficients, the numbers you find in Pascal's Triangle, which tell us how many ways we can pick a group of items from a larger pile. For decades, mathematicians have been fascinated by a simple question: if you pick a number, say 6, how often does it divide these combinations? A famous result by Singmaster in 1974 revealed a surprising truth: almost every single binomial coefficient is divisible by any given number. It's as if, in a massive lottery, almost every ticket is a winner for the number 6. But mathematicians are naturally curious about the losers—the tiny, rare exceptions that aren't divisible. While we know exactly how these exceptions behave when the divisor is a single prime number (like 2 or 3), the picture gets foggy and mysterious when the divisor is a mix of primes, like 6. Understanding these rare, stubborn numbers is like finding the few grains of sand that refuse to dissolve in the ocean; it helps us understand the deep, hidden structure of numbers themselves.

In this paper, Pascal Jelinek tackles the mystery of the number 6. Specifically, he looks at the binomial coefficients that are "coprime" to 6, meaning they share no common factors with 6 at all. Since 6 is made of the primes 2 and 3, a number is coprime to 6 only if it is not divisible by 2 (it's odd) and not divisible by 3. The author starts with a "trivial" estimate, which is a rough guess based on counting all the odd numbers in the triangle. However, this guess is too generous; it assumes that being odd and avoiding 3 are completely independent events, like flipping two separate coins. Jelinek proves that this isn't quite true. By using a clever mathematical trick involving the "digits" of numbers in different bases (specifically base 2 and base 3) and applying a theorem by Stein about how these digits interact, he shows that the two conditions actually restrict each other.

The result is a significant improvement on our understanding. Jelinek demonstrates that the number of these special coefficients is actually smaller than the rough guess suggested. He provides a new, tighter upper bound, showing that the count of these numbers is roughly the size of the odd numbers divided by a factor that grows with the square of the logarithm of the total size. In simpler terms, he proves that these "coprime to 6" numbers are even rarer than we previously thought, shaving off a significant chunk of the estimated count. This is the first time a non-trivial, rigorous bound has been established for this specific case, moving the field from a vague guess to a more precise mathematical reality. The paper doesn't just say "it's smaller"; it gives a specific formula showing just how much smaller, offering a new tool for understanding the intricate dance of divisibility in the number system.