Twin Primes: The Mystery of Primes That Travel in Pairs
3 and 5. 11 and 13. 17 and 19. 41 and 43. Primes that differ by exactly 2 are called twin primes, and they keep appearing no matter how far you search. The largest known pair has 388,342 digits.
Are there infinitely many? It's one of the oldest open questions in mathematics, and nobody knows. But in 2013, a mathematician who had spent years working in obscurity, at one point reportedly keeping books for a Subway sandwich franchise, proved a result that stunned the field and brought the answer closer than ever.
The Sum of Their Reciprocals Is Finite
Euclid proved there are infinitely many primes over 2,000 years ago. Euler went further: the sum of the reciprocals of all primes, 1/2 + 1/3 + 1/5 + 1/7 + …, diverges to infinity.
You might expect the same for twin primes. It isn't. In 1919, Norwegian mathematician Viggo Brun proved the sum of reciprocals of twin primes converges:
(1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + … ≈ 1.902
That value is called Brun's constant. Surprisingly, this result doesn't settle whether twin primes are infinite. A convergent sum can have finitely or infinitely many terms. It only tells us twin primes are much rarer than primes in general.
The Basics
A twin prime pair is (p, p + 2) where both are prime. The first few:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), …
Some quick facts:
- There are 35 twin prime pairs below 1,000 and 8,169 below 1,000,000.
- Every twin prime pair after (3, 5) has the form (6k − 1, 6k + 1). The number between them is always a multiple of 6.
- (3, 5, 7) is the only prime triple with gaps of 2. Among any three numbers n, n + 2, n + 4, one is divisible by 3.
Find your own pairs in the list of prime numbers.
The Twin Prime Conjecture
The twin prime conjecture says there are infinitely many twin prime pairs. It's often associated with Alphonse de Polignac, who proposed a more general version in 1849: for every even number k, there are infinitely many pairs of consecutive primes that differ by k.
In 1923, G. H. Hardy and John Littlewood proposed a precise estimate of how many twin primes there should be up to x:
π₂(x) ≈ 2C₂ × x / (ln x)², where C₂ ≈ 0.6602
Computer counts match this prediction remarkably well. Everyone believes the conjecture is true. Nobody has proved it. Learn more about Hardy in our post on G. H. Hardy.
An Insider Reference: Yitang Zhang's Breakthrough
In April 2013, Yitang Zhang, then a little-known lecturer at the University of New Hampshire, submitted a paper to the Annals of Mathematics. After finishing his PhD in 1991, he had struggled to find an academic job and worked for years outside academia, including stints as an accountant and at a Subway restaurant.
His paper proved that there are infinitely many pairs of primes that differ by at most 70,000,000.
That might sound like a huge gap. But before Zhang, nobody could prove that prime gaps stay below any fixed number infinitely often. Going from "no bound at all" to "70 million" was the leap. Getting from 70 million down to 2 is "just" a matter of improving the number.
The Race to Shrink the Gap
The mathematical community responded fast:
| Date | Bound on gaps occurring infinitely often | Who |
|---|---|---|
| April 2013 | 70,000,000 | Yitang Zhang |
| mid-2013 | 4,680 | Polymath8 collaborative project |
| November 2013 | 600 | James Maynard (independently, simpler method) |
| 2014 | 246 | Polymath8b, building on Maynard |
The Polymath project was an online collaboration organized by Terence Tao, with many mathematicians refining the argument in public blog comments. James Maynard received the Fields Medal in 2022, partly for this work.
Under an unproven assumption called the generalized Elliott–Halberstam conjecture, the methods reach a gap of 6. Getting to 2 appears to need a genuinely new idea.
The Pentium Bug
In 1994, Thomas Nicely, a mathematics professor at Lynchburg College in Virginia, was computing Brun's constant using many PCs. His results didn't agree across machines.
He tracked the discrepancy to a flaw in Intel's new Pentium processor: its floating-point division gave wrong answers for certain rare inputs. The "Pentium FDIV bug" became a public scandal, and Intel eventually offered replacement chips, taking a charge reported at $475 million. A search for twin primes had exposed a hardware defect in one of the world's most popular chips.
Two Concepts Worth Knowing
Prime Gaps
A prime gap is the difference between consecutive primes. The average gap near x is about ln(x), but gaps vary wildly. Twin primes are gaps of 2, the smallest possible after (2, 3).
Convergent Series
A convergent series adds up infinitely many terms to a finite total. Brun's theorem shows the reciprocals of twin primes form one, unlike the reciprocals of all primes.
Quick Answer: Are There Infinitely Many Twin Primes?
Nobody knows. Twin primes are pairs of primes that differ by 2, like 11 and 13. The twin prime conjecture says there are infinitely many, but it hasn't been proved. In 2013, Yitang Zhang proved infinitely many prime pairs differ by at most 70 million, and later work reduced that bound to 246.
Try Them Yourself
- List of Prime Numbers: spot twin pairs yourself
- Prime Checker: test whether p and p + 2 are both prime
- Is 101 Prime?: check one half of the twin pair (101, 103)
- Number 6: the number sitting between twin primes
- Logarithm Calculator: estimate twin prime counts with Hardy–Littlewood
- Why There Are Infinitely Many Primes: the proof twin primes still lack
Count the twin prime pairs below 100 (there are 8), then below 200. Compare your counts with the Hardy–Littlewood estimate and see how close it gets.