Artin’s Conjecture: Why ~37% of Primes Have Maximum Period in 1/p
“I was careless, but the machine caught up with me.” This quote belongs to Emil Artin (pictured), one of the founders of modern abstract algebra. In 1957, Derrick and Emma Lehmer, working with early computers, discovered a discrepancy in an estimate Artin had proposed back in 1927 — and that estimate became known as Artin’s conjecture.
What Artin’s Conjecture Says About Fractions 1/p
Recall that fractions of the form 1/n are either finite or infinite but repeating — consisting of cycles. The maximum possible length of each cycle equals n − 1. For some values of n this maximum is achieved, for others it is not: 1/7 achieves it, while 1/13 does not.
This raises a natural question: how many primes p are there for which 1/p has a cycle of maximum length?
That is precisely the question Artin’s conjecture answers. And the answer is: infinitely many.
The ~37% Estimate at the Heart of Artin’s Conjecture
Beyond the mere infinitude, Artin went further and heuristically computed the proportion of all primes for which the cycle length is maximal. His estimate: roughly 37%.
But what makes this even more striking is not the number itself — it’s the fact that nearly 100 years later, Artin’s conjecture is still a conjecture. The proof has turned out to be extraordinarily difficult.
Partial Results: Hooley, Heath-Brown, and the Riemann Hypothesis
In 1967, Christopher Hooley proved Artin’s conjecture — but only conditionally, assuming the Generalized Riemann Hypothesis holds. And the Generalized Riemann Hypothesis is itself one of the greatest unsolved problems in all of mathematics.
In 1986, Heath-Brown proved an unconditional version, but only in a partial case. He showed that at least one of the numbers 2, 3, or 5 has the property that the period of 1/p is maximal for infinitely many primes p. The catch: we cannot say which of the three it actually is.
Why Artin’s Conjecture Matters for Number Theory
What began as a seemingly elementary puzzle about simple fractions of the form 1/n turns out to be one of the deepest open problems in number theory — and one entangled with the Riemann Hypothesis itself.
This gives a concrete answer to anyone who asks why the Riemann Hypothesis is worth proving: one compelling reason is that its proof would immediately yield a proof of Artin’s conjecture.
And there is a broader lesson here. The beauty of 1/n lies in its deceptive simplicity. What looks harmless and transparent sits right next to the central open problems of number theory.
