What Don’t Mathematicians Know About Prime Numbers?

Some numbers split in only one way: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 12 = 5 + 7. Others have several. Here is every way for the even numbers up to 40, counting 3 + 7 and 7 + 3 as the same way:

Even number	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40
Ways	1	1	1	2	1	2	2	2	2	3	3	3	2	3	2	4	4	2	3

Look at the table, then use the widget below to look much further.

• a.Does the number of ways grow steadily, or does it jump about?
• b.30 has 3 ways but 32 has only 2; 36 has 4 but 38 has 2. What do 30 and 36 have in common?
• c.Up to 10,000, does the smallest number of ways stay near 1, or does it climb?

Why multiples of 3 have twice the ways

Take 1,000, which isn’t a multiple of 3. Try a prime p from 5 upward. Every prime from 5 on leaves 1 or 2 when divided by 3. If p leaves the same remainder as 1,000 (which leaves 1), then 1{,}000 − p leaves 0: it is a multiple of 3, so it can’t be prime. That kills about half the primes before you even look: of the 93 primes from 5 to 500, 45 give a multiple of 3.

Now take 1,002, a multiple of 3. Then 1{,}002 − p is a multiple of 3 only if p is, and no prime from 5 on is. None of the 93 primes is ruled out this way. So a multiple of 3 has about twice as many chances, and gets about twice as many ways. The same thing happens, more weakly, with 5 and 7: 210 = 2 × 3 × 5 × 7 has 19 ways, while 212 has only 6.

On 7 June 1742, Christian Goldbach, a mathematician in the service of the Russian government, wrote to his friend Leonhard Euler about numbers made by adding primes. In that letter and Euler’s reply, the claim took the shape we know:

A conjecture is a statement that mathematicians think is true but nobody has proved. Once proved, it becomes a theorem. Euler answered on 30 June 1742, in German: “That every even number is a sum of two primes I hold to be a completely certain theorem, although I cannot prove it.” (Our translation.) In those days 1 counted as a prime, so his letters say things a little differently, but this is the modern form.

How sure can anyone be?

The evidence from computers. In April 2012, a search led by Tomás Oliveira e Silva at the University of Aveiro, in Portugal, run in the spare time of many computers, finished checking every even number up to 4 × 10¹⁸. Every one splits. None of them needs a smaller part bigger than the prime 9,781.

The evidence from counting. Near a number n, about one number in every $\ln n$ is prime, where $\ln n$ is the natural logarithm of topic 3, about 2.3 for every factor of 10 in n. (Near 10,000, about one number in 9.2.) To split n you have about n/2 pairs to try, and if each part were prime by chance, a pair would be two primes about once in every $(\ln n)^2$ tries. So you’d expect roughly

$$\frac{n}{2(\ln n)^2}$$

ways. For 10,000 that gives about 59. The true count, 127, is higher, because the formula forgets what you just saw: both parts are odd, and multiples of 3 and 5 aren’t ruled out at random. But the formula gets the main thing right: the ways grow without limit. Near 4 × 10¹⁸ it predicts about a thousand million million ways. For a number to have none, every one of those chances would have to fail at once.

The estimate that was always too high, until it wasn’t

Here is a sharper warning, about counting primes themselves. In 1792 or 1793, aged 15 or 16, Carl Friedrich Gauss thought about how many primes there are up to x. The estimate that grew from it adds up the chance $1/\ln t$ of being prime for every number t up to x (smoothly, as calculus adds). That sum is written $\mathrm{li}(x)$. It is remarkably good:

Up to	Primes there are	Gauss’s estimate, li	Too high by
1,000	168	177.6	9.6
1,000,000	78,498	78,627.5	129.5
10⁹	50,847,534	50,849,235.0	1,701.0
10¹⁸	24,739,954,287,740,860	24,739,954,309,690,415.0	21,949,555.0

Every count anyone has ever done finds the estimate too high. So surely it always is? In 1914 John Littlewood proved it is not: somewhere, the count of primes overtakes Gauss’s estimate, and then they swap places infinitely often. A crossing has been proved near 1.4 × 10³¹⁶, a number with 317 digits, far beyond any computer’s count; none comes before 10¹⁹, and exactly where the first is, nobody knows. (That was the first job of Skewes’s number, in topic 5.) A pattern can hold for every number anyone can check and still be false.

Why it is so hard

Primes are defined by multiplying: a prime isn’t a product of smaller numbers. Goldbach asks about adding them, and knowing how a number breaks into factors tells you almost nothing about the factors of its neighbors. The tools for crossing between the two worlds are still too weak.

What has been proved

Mathematicians have proved things that come close:

• Three primes for odd numbers. Goldbach’s other guess in the 1742 letter, written in its margin, in modern form: every odd number greater than 5 is the sum of three primes (35 = 2 + 2 + 31, 99 = 3 + 7 + 89, 2,025 = 3 + 5 + 2,017). Ivan Vinogradov proved it in 1937 for all odd numbers beyond some huge size; Harald Helfgott proved it for all of them in 2013 (accepted for publication in 2015, and still being revised chapter by chapter).
• A prime plus almost a prime. In 1973 Chen Jingrun proved that every large enough even number is a prime plus a number that is either prime or the product of two primes: 2,026 = 5 + 2,021, and 2,021 = 43 × 47.
• Almost every even number. In 1937 and 1938 Nikolai Chudakov, Johannes van der Corput and Theodor Estermann proved that exceptions, if there are any, become rarer and rarer: as you go further out, the share of even numbers that can’t be split shrinks towards zero. In 1975 Hugh Montgomery and Robert Vaughan proved that they are rarer still.

Primes come close together surprisingly often: 11 and 13, 17 and 19, 29 and 31, 101 and 103. Two primes that differ by 2 are twin primes. (Apart from 2 and 3, they can’t be closer: of two numbers 1 apart, one is even, and the only even prime is 2.)

Here is how many twin pairs there are, counted by our program up to 10⁸ and by others beyond:

Up to	100	1,000	10,000	100,000	10⁶	10⁷	10⁸	10¹⁸
Twin pairs	8	35	205	1,224	8,169	58,980	440,312	808,675,888,577,436

They thin out, but they keep coming. The largest twin pair known (as of October 2026) is 2,996,863,034,895 × 2¹²⁹⁰⁰⁰⁰ ± 1, two numbers of 388,342 digits found in September 2016 by the volunteers of the PrimeGrid project. G. H. Hardy and John Littlewood guessed a formula for how many there should be; up to 10⁸ it predicts about 440,368, against the true 440,312.

It is one of the best-known open questions in mathematics: the counting evidence says twins never stop, and nobody has proved it.

A gap that stays small

Before 2013 nobody could prove even that the gaps between neighboring primes stay below some fixed size infinitely often. Then, in April 2013, a university lecturer, Yitang Zhang, proved it: there are infinitely many pairs of primes less than 70 million apart. Seventy million is not two, but it is a fixed number, and that was the breakthrough. Within weeks, mathematicians around the world, working together online in Terence Tao’s Polymath project, began cutting it down. In November 2013 James Maynard, with a new and simpler method, reached 600. By 2014 the Polymath project, using Maynard’s ideas, had 246.

In Figure 10.2, from 70 million to 246 the bound fell by a factor of about 285,000; from 246 to 2 it needs a factor of only 123. But the best these methods give, even assuming a stronger unproved guess about how primes spread, is 6. And “246” means only that at least one of the gaps 2, 4, 6, …, 246 happens infinitely often. Nobody knows which.

In 1912, at the International Congress of Mathematicians, the German mathematician Edmund Landau listed four questions about primes that he called “unattackable at the present state of mathematics”. Two you have met. As of 2026, all four are still open.

1. Goldbach’s conjecture: every even number greater than 2 is a sum of two primes.
2. Twin primes: infinitely many pairs of primes differ by 2.
3. Legendre’s conjecture: between any two square numbers next to each other there is always a prime.
4. Primes one more than a square: there are infinitely many primes of the form $n^2 + 1$.

A prime between every two squares?

Between 1 and 4 there are 2 and 3; between 4 and 9, 5 and 7; between 9 and 16, 11 and 13; between 16 and 25, 17, 19 and 23. Adrien-Marie Legendre guessed, in his book on number theory (1808), that this never fails.

The counts grow: for every n from 1,000 to 9,999 there are at least 127 primes between $n^2$ and $(n + 1)^2$, and the conjecture has been checked up to n of about 4.5 × 10⁹. What is proved, Bertrand’s postulate (Chebyshev, 1852), promises a prime between any number and its double: from 10,000, somewhere in a stretch of 10,000 numbers. Legendre asks for one in a stretch of only 201, from 10,000 to 10,201. Between two big enough cubes next to each other a prime has been proved, but squares are still too close together for the proofs.

Primes one more than a square

$n^2 + 1$ is prime for n = 1, 2, 4, 6, 10, 14, 16, 20, …: the primes 2, 5, 17, 37, 101, 197, 257, 401. Among n up to 100 there are 19 such primes; up to 10,000, 841. They grow rarer, but nobody can prove they don’t stop. The nearest result: in 1978 Henryk Iwaniec proved that infinitely many numbers $n^2 + 1$ are a prime or the product of two primes, close in spirit to Chen’s theorem.

Gauss’s estimate $\mathrm{li}(x)$ missed by 9.6 at 1,000 and by about 22 million at 10¹⁸. That is tiny next to the counts, and smaller even than $\sqrt{x}$, the square root: at 10¹⁸, $\sqrt{x}$ is 10⁹, and the miss is about 2 % of it.

In 1859 Bernhard Riemann linked the count of primes to a single function and the places where it is zero, and guessed that all those places, apart from some easy ones, lie on one straight line. His guess, the Riemann hypothesis, comes down to this: Gauss’s estimate is never wrong by much more than $\sqrt{x}$. If it is true, the miss is always less than $\sqrt{x} \times \ln x \div 8\pi$ once x passes 2,657 (Lowell Schoenfeld, 1976); at 10¹⁸ that allows about 1.6 billion, and the real miss is 22 million. The two go together: if the miss always stayed that small, the hypothesis would be true. So it says that the primes, scattered as they look, are spread as evenly as they possibly could be (in 1901 Helge von Koch showed that it gives what mathematicians call the best possible bound on the miss).

In 1900 David Hilbert put it, together with Goldbach and the twin primes, in his famous list of 23 problems for the new century. In 2000 the Clay Mathematics Institute named it one of seven Millennium Prize Problems, with US$1 million for a proof. Of the seven, only one has been solved; the Riemann hypothesis hasn’t.

Most of these questions have no use outside mathematics yet. But one practical cousin was answered recently, in South Asia.

• Is There a Biggest Prime Number? (Grades 6–8) Euclid’s proof that the primes never end, and how they thin out.
• Is a Pattern That Works Ten Times Always True? (Grades 6–8) Goldbach’s other guess failing at 5,777, and patterns that break late.
• What Is the Biggest Number Anyone Has Used? Skewes’s number, where the count of primes and Gauss’s estimate cross.
• How Can Two Strangers Agree on a Secret in Public? Big primes at work, keeping your messages private.
• Can You Prove √2 Is Not a Fraction? What a proof is, and why checking cases is never one.
• Computing, Going Further, Grades 9–12: Will This Simple Rule Ever Stop? Another question a child can ask and no one can answer, the Collatz problem, explored by program. (Coming soon.)