What it all means: the open questions in plain words

Five new prime fractions for pi. Each fraction is built from the two before it, so the bottom numbers grow fast: by step 25,332 the bottom number has 13,017 digits. Teddy Rook tested them one by one with his own programs, top and bottom numbers, to step 50,000, and found that the bottom numbers at steps 25,332, 27,651, 32,394 and 37,721 are probable primes, and so is the top number at step 42,037. To go faster, he first threw out every candidate with a small factor, using a “rhythm sieve”: each small prime is a voice that sounds on the steps where it divides the bottom number, and only the steps where every voice is silent need the slow test. You can hear it on the Rhythm of Numbers page. We rebuilt all five numbers from scratch and repeated the search ourselves, the bottom numbers to step 38,571 and the top numbers to step 42,037: there are no others there. The new primes’ page shows all the digits and a short program to check them.

Primes in batches. Between two of pi’s best fractions lie the in-between fractions. Where pi has a big term, there is a long run of them, all of about the same size, and a single sieve serves the whole run. Primes turn up in batches there: the first run gives 29, 43 and 71 (Archimedes’ 223/71 is one of these fractions, and so is the paper’s opening guess, 3 + k/(7k + 1). Written as (22k + 3)/(7k + 1), that guess makes a whole family of fractions; 223/71 is the first with a prime on top and on the bottom, and then come 619/197, 751/239, 883/281 and many more. Whether that goes on forever is unknown, just as for twin primes), the run belonging to the famous 292 gives 33 primes of the form 106 + 113k, and up to step 3,000 there are 111 such primes against 7 prime best fractions. These are not the fractions the open question asks about, and each one is only as likely to be prime as any number of its size; there are simply many more of them.

e’s pattern, pinned down. e = 2.71828… has a continued fraction with a pattern: 2; 1, 2, 1, 1, 4, 1, 1, 6, …. Divide its fractions’ bottom numbers by a prime, say 7, and look at the remainders. We already knew they repeat. Teddy Rook’s notes prove exactly how: after 3 × 7 = 21 steps the remainders come back with the opposite sign, and after 42 steps they come back exactly. The same holds for every prime. They also show that every whole number m divides three of the bottom numbers, so two steps out of every three can never give a prime. All ten prime bottom numbers of e known so far sit on the remaining steps. The tool is a set of exact formulas for every fraction of e, from integrals that Charles Hermite wrote down in 1873.

What the new primes say. The paper’s rule of thumb has two parts: the odds that a number of that size is prime, and a correction, because a fraction’s top and bottom never share a factor, which makes its bottom number a little more likely to be prime (by a factor π²/6, about 1.64). Until now pi’s own primes were too few to tell whether the correction applies to pi. With the five new ones they can: pi’s count of prime fractions now favors the corrected rule over the uncorrected one by about 30 to 1. A second check uses the rhythm sieve’s “voices”: over two million of pi’s fractions, every prime below 1,000 divides the bottom numbers at its proper rate (2 divides a third of them, 3 a quarter, and so on), just as for a typical number. Both are evidence that pi behaves like a typical number, not proof.

The correlation matrix’s growth. Its largest value grows with the size of the table, but more slowly than any power of the size and faster than any power of its logarithm. Teddy Rook’s notes find the exact rate, up to a correction that fades away, and show that the smallest value shrinks toward zero at exactly the mirror-image rate. The method is a known one from the study of the zeta function, so experts may see this as a natural consequence of their tools; we did not find it written down for this table. His notes also prove that the crowd of values settles into a fixed shape as the table grows. We know four exact numbers that describe that shape (1, 3, 0.5599 and 3.2796), and the famous 6/π², which the paper asked about, is not one of them.

In math, a proof settles a question for good. Computer results, however many, can only support a guess. The paper labels every claim one way or the other, and so does this page.

A bunch after a long wait. The 10th prime fraction is at step 6,925, then nothing until step 25,332, and then four in quick succession: 25,332, 27,651, 32,394 and 37,721. By the rule of thumb, a typical number would put four in that stretch less than 1% of the time. But we picked the stretch after seeing the primes, which makes any stretch look special, and the stretch before it ran low (10 found by step 20,000 where about 13 were expected). Over the whole range the total is right on: 14 against about 14.1 by step 50,000, and none at all from 40,000 to 50,000. The top numbers did the opposite: nothing from step 8,750 to 42,037, a wait that long happens about 1 time in 9. Primes that arrive at random often clump like this. The chance of a 15th prime bottom number by step 62,500 is about 27%, by step 100,000 about 62%, and by step 200,000 about 85%.

How rare these primes are. Big numbers are rarely prime: among numbers with 13,000 digits, only about 1 in 30,000 is prime, and pi’s bottom numbers do a little better, about 1 in 18,000. Pi’s bottom numbers grow by about one digit every two steps, so the chance keeps dropping. Add it all up and the expected number of prime fractions grows like a logarithm: roughly one more each time you double the number of steps. Think of a lottery where the tickets get worse every week: the wins keep coming, but further and further apart.

The long gap. The 10th prime fraction is at step 6,925 and the 11th at step 25,332, almost four times as far. That looks strange, but the rule of thumb expected only about 1.8 primes in that stretch, and gave about a 1 in 6 chance of none at all. Long gaps are normal, and they will get longer. To expect about two more, you would have to search out to step 100,000, where the numbers have about 51,000 digits.

A lean toward G. The paper turns a prime into a note by its remainder after dividing by 12, and every prime lands on one of four notes: C♯, F, G or B (together they make a chord). Pi’s fourteen prime bottom numbers play the motif G F C♯ G G G G G C♯ F B F G B. Add the 16 prime top numbers and G gets 15 of the 30. If the four notes were equally likely, that would happen less than 1% of the time. Still, it looks like luck. The bottom numbers that could be prime land on the four notes almost exactly equally, more than 4,000 times each up to step 33,000. Only the ones that turned out prime lean. The primes among the in-between fractions show no lean toward G either. And we spotted the lean after looking at many things, so something was bound to stand out. The three primes found since we noticed it are a G, a B and an F (the new top number). If it is luck, about a quarter of the next prime fractions will be G. You could play the motif, though.

e is not typical. e’s fractions are divisible by small primes about twice as often as random numbers. Half of that is now explained: three out of every 3ℓ of them are forced to be divisible by the prime ℓ, and chance seems to supply about three more.

Modest ones. This is pure math. No bridge or bank depends on whether pi has infinitely many prime fractions.

It adds one more piece of evidence to the picture that pi behaves like a “random” or typical number, a picture that many experts believe and no one can prove.

It shows how math research often works now: computer experiments suggest what is true, and then people try to prove it. The rule of thumb for prime fractions came from experiments like these, and for almost every number it is now a theorem.

It is also a nice example of how someone outside a university, with a good computer and careful checks, can add a new entry to a list that mathematicians have kept for years.

The proofs. A certificate that q₂₅₃₃₂ is prime is being computed. It may take a week or more. When it is done, the new primes’ page will say so. The 12th would take roughly 40% longer, the 13th about 2.7 times as long, the 14th about 5 times and the new top number about 8 times as long; none of those has been started.

Searching further. Each step costs more than the last, because the numbers keep growing. The next prime fraction of pi may be tens of thousands of steps away. Testing the G candidates first would cost nothing, but skipping the others could miss the next prime, so the search has to test them all.

Proving it for pi. Showing that pi itself has infinitely many prime fractions, or that pi is typical, is far beyond today’s methods. The paper proves the rule for almost every number, and that is likely as far as anyone can go for now.