We look for prime number patterns in the seeds of a sunflower

1Twins are also more even than chance

The firmest: a formula that comes out of the computation, is right to three digits, and that we did not find published.

• Idea already known
• Needs more checking

What you see

We repeated with pairs of twin primes the measurement we made with primes: how much their number changes from one ring to the next. They also vary less than chance, but far less markedly: between 71% and 90% of the variation of chance depending on the stretch, from ring 300 to ring 3,000,000, rising slowly with the radius: 81% near ring one million. For pairs at distance 6 it is around 86%.

Why it happens

It comes out of the Hardy–Littlewood formulas applied to four primes at once: two pairs of twins cannot sit side by side (of three consecutive odd numbers one is a multiple of 3), and that repulsion spreads them out better than a lottery would. We computed the prediction and it goes from 71% at ring 450 to 81% at ring 1,500,000; what we measured on eight stretches stays within 5 points of it. The formula above sums up that computation: its three numbers come from fitting the Hardy–Littlewood sum, not from the data. The two toughest tests: on the 39,000 rings from 1,000 to 40,000, the measured deviation is 0.999 of what the formula gives and unusual rings show up as often as a bell curve says; and 26 stretches around ring one million gave 0.806 ± 0.007, where the formula predicts 0.807. For pairs at distance 6 one more term is needed, because three primes can stand in a row (p, p + 6, p + 12): the prediction is 0.86 and we measured 0.859 ± 0.011. According to the computation, the variation keeps rising very slowly and levels off near 88%, perhaps exactly 7/8, for pairs at any distance (we computed 2, 4, 6 and 30); that of primes, by contrast, tends to one half. That limit is what still needs checking: it is too far away to measure. If the interval has length x to the power δ, it points to a rule: the variation tends to 1 − δ for primes, which is exactly what the Montgomery–Soundararajan paper says, and to 1 − δ²/2 for pairs, which we did not find published (we searched, though not exhaustively). For prime triples the numerical computation is not enough to decide whether the pattern continues (it would be 1 − δ³/6), although an argument about how divisors add up suggests it does. The same computation works for other patterns: pairs at distance 4 and two kinds of triples, whose measured variation (0.92 to 0.96) lies within the error of the prediction (0.94). When auditing the finding we redid the sum by a different route and got the same.

2Along the curves of the Sacks spiral, primes are also more even than chance

A prediction with nothing fitted that matches the measurements and that we did not find published either; it lacks a closed formula.

• Idea already known
• Needs more checking

What you see

Each curve of the Sacks spiral is a polynomial n² + k, and the Bateman–Horn conjecture says how many primes it should have. We measured how far each curve strays from that count. Counting primes with n between N and 2N, the variation is 42% of that of chance, and it does not change with size: it is the same from N = 3,000 to N = 300,000. It does change with the length of the stretch: 47% from N to 1.25 N, 37% from N to 4N and 31% from N to 32N.

Why it happens

We computed it from the Hardy–Littlewood formulas, without fitting any number. Two values on the same curve, n² + k and m² + k, lie m² − n² apart; those formulas say how likely both are to be prime depending on the divisors of that distance. Summing over all pairs, the dominant part cancels exactly with what each curve's constant already explains, and the remainder, which is negative, is the regularity. The prediction gives 40 to 43% for stretches from N to 2N (we measured 41 to 43%) and drops as the stretch lengthens just as the measurement does: 47, 45, 37, 36, 33 and 33% where we measured 47, 44, 37, 34, 33 and 31%, with an error of one point on each measurement. In a first comparison the prediction sat 2 to 4 points lower; that was our fault, because we averaged the curves one way when measuring and another when predicting. With the same average the difference disappears. We searched for whether this was published and did not find it: what exists are results about the average (almost every curve has the expected primes), not about how far they stray. We have no closed formula, only the sum. We repeated everything with new code and with two other families, n² + n + k and 2n² + k: in all three the average prediction matches the measurement to within one point (41, 38 and 40%). Evaluated on an exact stretch the prediction wobbles by 1 to 3 points depending on the endpoints, and we do not fully understand that.

3A formula predicts how even the rings are

The basis for all the others, checked up to numbers around ten trillion; the idea was already published.

• Idea already known

What you see

The number of primes changes from one ring to the next much less than if they fell at random. That variation, as a fraction of chance, follows the formula ½ − 1.054 / ln n. We measured it first: the fit on rings up to 10,000 gave 1.05. Then we tested it on eleven separate stretches between ring 100,000 and ring 3,000,000, and it held in ten. It also works for other widths: for an interval of width λ·n the constant is ½ (ln λ + γ + ln 2π − 1), which grows by 0.35 every time the width doubles; we measured 0.33.

Why it happens

It is what a known result about primes in short intervals predicts (Montgomery and Soundararajan, 2004). We checked it against the original paper: its Theorem 3 gives the variation at each point with the constant 1 − γ − ln 2π, and that is where the 1.054 for rings comes from. At first it did not add up: we got 1.55. The mistake was ours, because we were using the version averaged from zero, which lacks that 1. Two caveats: the theorem rests on a strong form of the Hardy–Littlewood conjecture, which is unproved, and it covers intervals of width up to √N; rings are 2√N wide, right at the edge, so applying it there is an extrapolation that our measurements support. Our part is having watched it hold ring by ring. One of the eleven stretches, the one at ring one million, drifted; measuring 26 stretches around that ring gave an average constant of 1.15 ± 0.07, compatible with the theory, so it was an ordinary fluctuation. Nor does it depend on the interval starting at a square: shifting it by 1, 9, 49, 99 or 999 places, or by half a ring, the variation measured between rings 1,000 and 20,000 stays between 36% and 40%, where the formula gives 36% to 39%.

4Within a ring, the spirals are not independent

Uses both structures of the sunflower at once and the prediction is right to four digits; it follows from known formulas.

• Idea already known
• Needs more checking

What you see

We put the sunflower's two structures together: we counted the primes of each spiral inside each ring. Each spiral, on its own, varies from ring to ring half as much as chance (50%). The whole ring varies even less (36 to 39%), and the difference comes from how the spirals offset one another: when one has primes to spare, the spirals whose residue modulo 21 differs from its own by a multiple of 3 or 7 tend to fall short. The rest do not notice.

Why it happens

We computed it from the Hardy–Littlewood formula for prime pairs, without fitting any number, and it matches what we measured on rings 3,000 to 8,000. A single spiral: predicted 0.5039, measured 0.5037. Two spirals differing by a multiple of 3: predicted −0.0191, measured −0.0185. By a multiple of 7: −0.0325 and −0.0330. The rest: 0.0000 and −0.0004. Adding everything gives the variation of the ring: 0.376 predicted and 0.376 measured. Between rings 8,000 and 20,000 it matches just as well, except for the case of 7 (−0.029 predicted, −0.026 measured). The reason is that two primes at a distance that is a multiple of 3 show up together more often than average, and once the residues are fixed that advantage is already discounted: what remains is a repulsion. It is a consequence of known formulas; we did not search for whether this breakdown by spirals is published.

5How far each ring is from being empty

Turns the regularity into a measure of how far Legendre is from failing; it is an argument, not a proof.

• Idea already known
• Needs more checking

What you see

With the variation formula, the number of primes per ring behaves like a bell curve. We tested it on the 90,000 rings from 10,000 to 100,000: the measured deviation is 1.000 of the predicted one, and unusual rings show up exactly as often as the bell says (237 beyond 3 deviations against 243 expected; 5 beyond 4 against 5.7). Oppermann's half rings behave the same way. That lets us predict the worst ring of each stretch, within 1 to 6%, and measure the margin of each conjecture. Legendre: an empty ring would be 7 deviations away at ring 35, 20 at ring 1,000 and more than 400 at ring one million. Oppermann: 4, 13 and 284. Brocard, in its worst case: 12 and 31 for the first two.

Why it happens

If rings follow a bell curve, the probability of one being empty collapses: from 1 in a trillion at ring 35 to a number with 92 zeros at ring 1,000. It is an argument for why Legendre, Oppermann and Brocard should be true, not a proof: we checked the bell out to 4.5 deviations, and the conjectures live much farther out in the tail, where nobody has shown it still holds. Treating primes in short intervals as normally distributed was already a known idea.

6From ring 123 on, every ring has twin primes

An exact rule that is easy to state, but only checked up to ring 40,000.

• Idea already known
• Needs more checking

What you see

Only twelve rings contain no pair of twin primes: 1, 9, 19, 26, 27, 30, 34, 39, 49, 53, 77 and 122. From 123 to 40,000 we found no more. It is like Legendre's conjecture, but for twins. With the formula for their variation the margin can be measured: a ring without twins would be 2.3 deviations away at ring 122 (which is why it still happens), 4.4 at ring 1,000, 10 at ring 10,000 and 65 at ring one million.

Why it happens

Twin pairs per ring keep growing (about 14 at ring 1,000, about 3,500 at ring one million) and their number varies less than chance, so staying at zero becomes ever less likely. That ring 122 is the last one is not proved: it would imply there are infinitely many twin primes, one of Landau's problems. The list of rings without twins had already been observed; our part is the margin. The 122 belongs to the squares: if the interval is shifted by 1 to 99 places, the last ring without twins falls between 53 and 162, and there are 9 to 17 in all (checked up to ring 5,000). The general rule is that they stop appearing early, not the exact number.

7How far each even number is from having no decomposition

The margin is huge, but the variation rests on a fitted straight line with no law behind it.

• Idea already known
• Needs more checking

What you see

For every even number we count in how many ways it is a sum of two primes and compare it with the Hardy–Littlewood formula. The difference behaves like a very narrow bell curve: its variation is between 20% and 33% of that of chance, from even numbers around ten thousand to around three hundred million, and it grows slowly with size. That lets us measure Goldbach's margin: an even number with no decomposition at all would be 23 deviations away near 10,000, 55 near 100,000, 131 near one million, 327 near ten million and about 1,300 near three hundred million.

Why it happens

Even numbers have more and more decompositions (about 100 near ten thousand, about 29,000 near ten million) and their count strays little from the prediction, so zero keeps getting farther away. It is an argument for why Goldbach should be true, not a proof: the formula is a conjecture, we only checked the bell on what we measured and we have no law for how the variation grows. What we did do is put it to the test: from the measurements up to ten million we fitted a straight line and used it to predict 30% and 32% for one hundred and three hundred million; we measured 33% at both, within the measurement error (about 3 points). We also set apart the even numbers with few expected decompositions, the ones that could fail: they vary just one or two points more than the rest, so the margin does not change. The underlying idea was already known.

8Lucky numbers are even more even than primes

Only a measurement: we have no formula and no complete explanation.

• Needs more checking

What you see

Lucky numbers come from a sieve similar to that of the primes, but one that removes by position rather than by divisibility: 1, 3, 7, 9, 13, 15, 21… They have almost the same density as primes and almost as many twins. We measured how much their number changes from one ring to the next: around 27% of what chance would give, from ring 100 to ring 6,400, without rising with the radius. Primes, on the same rings, go from 30% to 38%. And no ring is left without lucky numbers: Legendre's conjecture also holds for them as far as we looked.

Why it happens

Regularity seems not to be exclusive to primes but something sieves produce: a sieve spreads its survivors better than a lottery. We put it to the test on the same rings. A lottery with the density of the primes gives between 93% and 101%, that is, no regularity at all. A sieve like that of the primes, but where each prime removes a randomly chosen residue class, gives between 24% and 32%: as even as the real primes. We have no formula for lucky numbers and found no published prediction, and the measurement uses a mean estimated from neighboring rings, not an exact one. One more clue towards that formula: we measured intervals of other widths, from a quarter of a ring to sixteen rings. The variation drops as the interval widens, as with primes but roughly one and a half times faster (from 48% to 11% near ring 4,500; primes go from 50% to 26%), and it barely changes with the radius.

1No ring is left without primes

It is Legendre's conjecture seen in the sunflower, and the medium findings grew out of it.

What you see

Squares land on integer radii. Between two consecutive radii there are always primes, and each ring holds more than the one before.

Why it happens

The radius of a seed is the square root of its number, so ring n holds the numbers between n² and (n+1)². It is Legendre's conjecture, drawn; Oppermann, Brocard and Andrica say the same with other widths. Squares are nothing special: if the interval is shifted by 1 to 99 places empty rings do appear, but only very small ones. The last is ring 6 shifted by 78 places, which fits entirely in the gap between the primes 113 and 127; from ring 7 to 5,000 there is none.

2A formula predicts how many twins each ring holds

Checks the classical prediction for twins, the tool we use in almost everything else.

What you see

We count the pairs of twin primes in the rings of the sunflower and compare them with the Hardy–Littlewood formula. In small rings the match is uneven (81%), and as they grow it settles at 100%. The same happens with Sophie Germain, n² + 1 and Goldbach.

Why it happens

The formula treats primes as if they were random with the right density and corrects for divisibility. Nobody has proved it: if it were true, there would be infinitely many twins. Matching ring by ring is evidence in favor, not a proof.

3Three spirals always lag behind

A real, measurable bias between spirals, explained for more than a century.

What you see

The 12 spirals that can hold primes do not get the same share. Those of residues 1, 4 and 16 are the three with the fewest primes up to one million, and on average they trail the other nine all the way up to 100 million.

Why it happens

This is Chebyshev's bias: primes slightly prefer residues that are not squares, and 1, 4 and 16 are exactly the squares modulo 21. It is a known phenomenon, explained assuming the generalized Riemann hypothesis; in the long run the twelve spirals share the primes equally. It happens with any number of spirals: we checked 3, 4, 5, 7, 8, 12, 13, 21, 30 and 34, and in all of them the non-square residues lead, between 96% and 100% of the time up to 100 million.

4At distance 144, primes pair up twice as often

Links the distances between sunflower neighbours to the prime-pair formula.

What you see

On the sunflower, the neighbors of a seed are a Fibonacci number away. We counted how many pairs of primes sit at each even Fibonacci distance, compared with twins: at distance 8 there are just as many, at 34 some 7% more, at 610 some 37% more and at 144 twice as many.

Why it happens

The Hardy–Littlewood formula predicts it: every odd prime q dividing the distance multiplies the pairs by (q − 1)/(q − 2). 144 is a multiple of 3 (factor 2) and 610 of 5 and 61 (factor 1.36). Up to 100 million, the counts match the formula with an error under 0.2%. That is why the traversal changes how close primes sit: each one makes numbers at different distances into neighbors.

5Sophie Germain primes fit in 6 spirals

Simple arithmetic: it is proved in a few lines.

What you see

The primes p for which 2p + 1 is also prime line up on just 6 rays.

Why it happens

Neither p nor 2p + 1 can be a multiple of 3 or 7. That leaves 5 residues modulo 21, plus the one of 3, which is the exception.

6Twin primes end up side by side

Simple arithmetic, from the same family as the previous one.

What you see

When only primes are written and each spiral is filled completely before moving on, twins show up as pairs of adjacent nodes.

Why it happens

Two twins are neighbors in the list of primes, and this traversal puts list neighbors on adjacent seeds of the same spiral.

7Primes live in only 12 of the 21 spirals

Explained by divisibility by 3 and 7 alone.

What you see

With the angle of 8/21 of a turn every spiral is a ray. Primes fill some rays and leave others empty, except for 3 and 7.

Why it happens

The spiral of a number depends on its residue modulo 21. A prime greater than 7 cannot be a multiple of 3 or 7, which rules out 9 of the 21 residues.

8How close primes sit depends on the traversal

A negative result: being close in the sunflower changes nothing.

What you see

With the “In order” traversal, each prime has its nearest prime 2% closer than in a control with the same density and the same residues modulo 42. With “Next” and with “Skip one” it is 1% farther.

Why it happens

On the sunflower, the neighbors of a seed are a Fibonacci number away, and each traversal turns those distances into different gaps between the numbers. Two primes show up together more or less often depending on whether their difference is a multiple of 5: once the control is also told to respect the residues modulo 5, the difference drops below one point in all three traversals (100.7, 100.1 and 100.4%), and with those of 11 as well, to half a point or less. We repeated the whole measurement when auditing the findings: the first time we had recorded 97, 101 and 100; the correct values are 97.7, 100.8 and 101.0. With the Lucas and silver angles the neighbors become Lucas and Pell numbers, and the effect drops to half a percentage point.