## March 19, 2016

### The Most Common Prime Gaps

#### Posted by John Baez Twin primes are much beloved. But a computer search has shown that among numbers less than a trillion, most common distance between successive primes is 6. It seems this goes on for quite a while longer…

… but Andrew Odlyzko, Michael Rubinstein and Marek Wolf have persuaded most experts that somewhere around $x = 1.7427 \times 10^{35}$, the most common gap between consecutive primes less than $x$ switches from 6 to 30:

• Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118.

This is a nice example of how you may need to explore very large numbers to understand the true behavior of primes.

They give a sophisticated heuristic argument for their claim—not a rigorous proof. But they also checked the basic idea using Maple’s ‘probable prime’ function. It takes work to check if a number is prime, but there’s a much faster way to check if it’s probably prime in a certain sense. Using this, they worked out the gaps between probable primes from $10^{30}$ and $10^{30}+10^7$. They found that there are 5278 gaps of size 6 and just 5060 of size 30. They also worked out the gaps between probable primes from $10^{40}$ and $10^{40}+10^7$. There were 3120 of size 6 and 3209 of size 30.

So, it seems that somewhere between $10^{30}$ and $10^{40}$, the number 30 replaces 6 as the most probable gap between successive primes!

Using the same heuristic argument, they argue that somewhere around $10^{450}$, the number 30 ceases to be the most probable gap. The number 210 replaces 30 as the champion—and reigns for an even longer time.

Furthermore, they argue that this pattern continues forever, with the main champions being the ‘primorials’:

$2$

$2 \cdot 3 = 6$

$2 \cdot 3 \cdot 5 = 30$

$2 \cdot 3 \cdot 5 \cdot 7 = 210$

$2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310$

etc.

Posted at March 19, 2016 6:10 AM UTC

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### Re: The Most Common Prime Gaps

Fantastic!

It seems you forgot a 5 in the last 2 primorials, though.

Posted by: Konrad Voelkel on March 19, 2016 10:35 AM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Posted by: John Baez on March 19, 2016 5:41 PM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

You meant “between $10^30$ and $10^30 + 10^7$” and “between $10^40$ and $10^40 + 10^7$”.

Posted by: Todd Trimble on March 19, 2016 12:20 PM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Posted by: John Baez on March 19, 2016 5:43 PM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

What about the simplest case – the race between twin primes and sexy primes?

I might be off by 1 or something, but with the help of a rudimentary script (in Matlab, of all things!), I believe that (541, 547) is the 26th sexy prime pair, and from then on there are more sexy primes than twin primes.

But this is not the first time the sexy primes overtake the twin primes – twin primes have the lead until (173,181) ties them at 12 before the twin primes retake the lead with (179,181). Later, sexy primes take the lead for the first time with the 22nd sexy prime pair (383,389) before it’s tied back up with (419, 421). There’s a bit more back-and-forth before sexy primes take the lead for good – or at least up to 1000, at which point there are 44 sexy primes and 35 twin primes and it seems unlikely that the twin primes will ever have the lead again.

I didn’t look at other prime gaps. It’s conceivable (but unlikely, I guess) that some other prime gap has the lead at some other point early on.

Posted by: Tim Campion on March 19, 2016 8:53 PM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Nice, Tim! Sometimes 4 has the lead over 2 and 6. Ian Stewart writes:

What number is the most common gap between successive primes less than $x$? This question was posed in the late 1970s by Harry Nelson of Lawrence Livermore National Laboratory. Later on, John Horton Conway of Princeton University coined the phrase “jumping champions” to describe these numbers.

The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. The sequence of gaps—the differences between each prime and the next—is 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2 and 4. The number 1 appears only once because all primes except for 2 are odd. The rest of the gaps are even numbers. In this sequence, 2 occurs six times, 4 occurs five times, and 6 occurs twice. So when $x$ = 50, the most common gap is 2, and this number is therefore the jumping champion.

Sometimes several gaps are equally common. For instance, when x = 5 the gaps are 1 and 2, and each occurs once. For higher $x$, the sole jumping champion is 2 until we reach $x$ = 101, when 2 and 4 are tied for the honor. After that, the jumping champion is either 2 or 4, or both, until $x$ = 179, when 2, 4 and 6 are involved in a three-way tie. At that point the challenge from 4 and 6 dies away, and 2 reigns supreme until $x$ = 379, where 2 is tied with 6. Above $x$ = 389 the jumping champion is mostly 6, occasionally tied with 2 or 4, or both. But when $x$ ranges from 491 to 541, the jumping champion reverts to 4. From $x$ = 947 onward the sole jumping champion is 6, and a computer search shows that this continues up to at least $x = 10^{12}$.

Posted by: John Baez on March 20, 2016 1:05 AM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

There go those small numbers, throwing a wrench in every pattern!

Posted by: Tim Campion on March 20, 2016 6:10 AM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Over on G+, Justen Robertson asked:

What’s the relationship between primorials and the point at which they become dominant?

I looked at the paper by Odlyzko, Rubinstein and Wolf, and near the start they give a rough formula for this They say the number

2⋅3⋅5 = 30

starts becoming more common as a gap between primes than

2⋅3 = 6

roughly when we reach

exp(2⋅3⋅4⋅3) = exp(72) ≈ 1.8 ⋅ $10^31$

That’s pretty rough, since they say actual turnover occurs around 1.7427 ⋅ $10^{35}$. But you probably can’t see the pattern yet, so let me go on!

The number

2⋅3⋅5⋅7 = 210

starts becoming more common as a gap between primes than

2⋅3⋅5 = 30

roughly when we reach

exp(2⋅3⋅5⋅6⋅5) = exp(900) ≈ $10^{390}$

Again, this is pretty rough - they must have a more accurate formula that they use elsewhere in the paper. But they mention this rough one early on.

I bet you still can’t see the pattern in that exponential, so let me do two more!

The number

2⋅3⋅5⋅7⋅11 = 2310

starts becoming more common as a gap between primes than

2⋅3⋅5⋅7 = 210

roughly when we reach

exp(2⋅3⋅5⋅7⋅10⋅9) = exp(18900)

The number

2⋅3⋅5⋅7⋅11⋅13 = 30030

starts becoming more common as a gap between primes than

2⋅3⋅5⋅7⋅11 = 2310

roughly when we reach

exp(2⋅3⋅5⋅7⋅11⋅12⋅11) = exp(304920)

Get the pattern?

Posted by: John Baez on March 20, 2016 1:13 AM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Leaves me wondering what happened to exp(primorial × 8 × 7) …

Posted by: Jesse C. McKeown on March 21, 2016 1:55 AM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Following wikipedia and using $p#$ to denote the primorial of $p$, I think the pattern is that $p_n #$ becomes more common than $p_{n-1}#$ at around $\exp(p_{n-1}# \cdot (p_n - 1) \cdot (p_n - 2))$. So the reason we don’t see $7# \cdot 8 \cdot 7$ is that 9 isn’t prime.

Posted by: Tim Campion on March 28, 2016 3:03 PM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

Yes, that’s the pattern, and the answer to Jesse’s question.

Posted by: John Baez on March 28, 2016 5:07 PM | Permalink | Reply to this

### Re: The Most Common Prime Gaps

The news seems to be trickling out. I got this email today:

PRIME NEWS: Last Digits 1,3,7 and 9 permeates significantly throughout primes whole complex.

Beeing the significant most frequent digits in primes, Last Digits 1,3,7 and 9 permeates throughout its whole complex.

The digits are not uniformly distributed but Benfordian, indicating primes are governed by a flux combination of Benfords Law and Last Digits. Hence consecutive LD=1 has greater probability. When sizing prime data-set, researchers should be aware of the importance of complete and fair rounds of first digits 1-9. Otherwise; the results may be biased.

You can see the numerical evidence, distributions and explanations in “primes” on www.stringotype.com.

It is hypothesized that the result corresponds to the dimensionless Fine Structure Constant - a fractal order in nature. Base 10 number system maps the order with zero skewness, giving rise to many circadian rythms beeing purely reflected in numbers. Hence digits corresponds to primary respondents and can possibly be intepreted as integrals of functional processes.

Best regards

Terje Dønvold

Oslo

Posted by: John Baez on March 30, 2016 7:54 PM | Permalink | Reply to this

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