❓ Still unsolved

What you’re seeing

Pick one of four problems at the top. Each has a box to try your own numbers and a chart.

• Twin primes. Type any number, up to $10^{18}$, and the page finds the next pair of primes two apart, like 11 and 13. The chart counts all twin pairs up to 2,000,000 (green) next to a prediction made in 1923 (gold).
• Goldbach. Type an even number and the page splits it into two primes. The chart, often called Goldbach’s comet, shows how many ways each even number up to 20,000 can be split. The number of ways grows, and the lowest edge of the comet never comes near zero.
• Collatz. Type a start and follow the halve-or-triple-plus-one rule down to 1. The Collatz page in this gallery explores this one in depth.
• Riemann hypothesis. Pick where to start on the line, and the page draws a wave called $Z(t)$. Each place it crosses zero (red dot) is a zero of the Riemann zeta function.

Every case you try will work. That’s the point: no one has found a counterexample, and no one has a proof.

Why it works

Testing huge numbers. To check whether a 15-digit number is prime, the page doesn’t try dividing by everything. It uses the Miller–Rabin test, based on Fermat’s little theorem: if $p$ is prime, then $a^{p-1} \equiv 1 \pmod p$. A composite number almost always fails some test of this kind. With the first 12 primes as the bases $a$, the test is known to give no wrong answers for numbers below about $3 \times 10^{24}$.

Twin primes. Primes near $x$ appear about once every $\ln x$ numbers. If two numbers two apart were prime independently, we’d expect about $x / (\ln x)^2$ twin pairs up to $x$. Hardy and Littlewood corrected this for the way divisibility by small primes affects neighbors, and predicted

$$\pi_2(x) \approx 2C_2 \int_2^x \frac{dt}{(\ln t)^2}, \qquad C_2 \approx 0.6601618.$$

The chart shows how closely real counts follow this.

Goldbach. An even number $n$ has many chances to be $p + q$: about $n / (2 \ln n)$ choices of prime $p$ below $n/2$, and each $n - p$ has about a $1/\ln n$ chance of being prime. That’s roughly $n / (\ln n)^2$ ways, which grows without bound. That’s why the comet rises.

The Riemann hypothesis. The zeta function starts as the sum

$$\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots,$$

and Bernhard Riemann showed in 1859 that its zeros control exactly how the primes are spread out. Besides “trivial” zeros at $-2, -4, -6, \ldots$, all its zeros lie in a strip, and Riemann guessed they all lie on the middle line where $s = \tfrac12 + it$. On that line there is a real-valued wave $Z(t)$, made by Hardy, with $|Z(t)| = |\zeta(\tfrac12 + it)|$. So every sign change of $Z$ is a zero on the line. The page computes $Z(t)$ with the Riemann–Siegel formula,

$$Z(t) \approx 2\sum_{n \le \sqrt{t/2\pi}} \frac{\cos\big(\theta(t) - t\ln n\big)}{\sqrt{n}},$$

plus a small correction term. The first zero is at $t \approx 14.1347$. The page also counts the zeros it finds and compares them with the number Riemann’s formula predicts. If a zero were ever off the line, there would be a missing sign change, and the counts would disagree.

Where it stands today

As of 2026, all four are unsolved. Here is what is proved and what has been checked.

Twin primes (are there infinitely many pairs $p$, $p + 2$?)

• In 2013 Yitang Zhang proved that infinitely many pairs of primes differ by less than 70,000,000. James Maynard (and, separately, Terence Tao) soon cut this to 600, and a large online collaboration (Polymath) brought it down to 246 in 2014. Getting from 246 to 2 needs new ideas.
• Assuming an unproved conjecture about how primes spread (the generalized Elliott–Halberstam conjecture), the gap would come down to 6.

Goldbach (is every even number above 2 a sum of two primes?) Christian Goldbach raised it in a letter to Leonhard Euler in 1742.

• Checked by computer for every even number up to $4 \times 10^{18}$ (Oliveira e Silva, Herzog and Pardi, 2014).
• The “weak” version, that every odd number above 5 is a sum of three primes, was proved by Harald Helfgott in 2013.
• Chen Jingrun proved in 1973 that every large enough even number is a prime plus a number with at most two prime factors.

Collatz (does every start reach 1?)

• Checked for every start up to $2^{71}$ (Bařina, 2025). Terence Tao proved in 2019 that almost every start falls very low, but not that every one reaches 1.

The Riemann hypothesis (are all the nontrivial zeros on the line $s = \tfrac12 + it$?)

• It is one of Hilbert’s 23 problems (1900) and one of the Clay Mathematics Institute’s seven Millennium Prize Problems (2000), with a \$1,000,000 prize.
• G. H. Hardy proved in 1914 that infinitely many zeros lie on the line. Since 1989, at least 40% of them are known to be on it.
• Every zero with $0 < t \le 3 \times 10^{12}$ has been checked and lies on the line (Platt and Trudgian, 2021).
• In 2024 Larry Guth and James Maynard proved a new limit on how many zeros could possibly be off the line, the first big improvement of its kind since the 1940s.

Try this

• Twin primes: search after 1,000,000,000,000. How far did you have to go? Try $10^{15}$. Do the pairs get rarer?
• Goldbach: try 100. Then try a few even numbers near 1,000,000. Is the smallest prime that works ever large?
• Goldbach: in the comet, find the stripes. Numbers divisible by 3 (like 30 or 60) sit higher. Why might that be?
• Riemann: start at 10 and find the first red dot. Then start at 1,000. Are the zeros closer together up there?
• Collatz: try any big number you like, up to 40 digits.