Unexpected biases in the distribution of consecutive primes
 

or [URL="https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/"]Mathematicians Discover Prime Conspiracy[/URL]

Paper: [url]http://arxiv.org/abs/1603.03720[/url]

Based on your first link I think prime numbers most likely in this order:

Prime numbers ending in 3 will occur most followed by 7. Prime numbers ending in 9 will occur more than primes ending in 1. I don't know this for sure but I know for a fact that most of the time there are more primes ending in 3 or 7 than in 1 or 9 in decimal. Is there a proof for this? Thanks to whoever knows how this is. :smile:

[QUOTE=PawnProver44;429037]Based on your first link I think prime numbers most likely in this order:

Prime numbers ending in 3 will occur most followed by 7. Prime numbers ending in 9 will occur more than primes ending in 1. I don't know this for sure but I know for a fact that most of the time there are more primes ending in 3 or 7 than in 1 or 9 in decimal. Is there a proof for this? Thanks to whoever knows how this is. :smile:[/QUOTE]

That's like saying "based on this link that says A is true, I therefore think this is most likely true: A".

:ban:

No one mentioned the frequency of primes ending in 1 or 9. Also I think the frequency of primes ending in 3 or 7 roughly have a tie. (Since 2 and 3 (mod 5) are nonresidues.)

[QUOTE=PawnProver44;429044]No one mentioned the frequency of primes ending in 1 or 9. Also I think the frequency of primes ending in 3 or 7 have a tie.[/QUOTE]

Did you read the paper or the article?

[QUOTE=axn;429031]or [URL="https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/"]Mathematicians Discover Prime Conspiracy[/URL]

Paper: [url]http://arxiv.org/abs/1603.03720[/url][/QUOTE]
Everything is partially controlled by whatever it favors or fights against. Even the resistance of a pattern is a pattern in itself.

I used to hate Rod Silverman until I realized that gave him a weird control over me.

Primes are a fun thing to seek patterns in because we think of them as resisting patterns, and you can do that with other things as well. Snowflakes are a good example, we think of them as being based on patterns, but really it's a sort of "fight" at the molecular level to NOT arrange themselves a certain way.

Edit: Just realized I sort of contradict myself above, but I'll leave it and hope people don't flame me. :)

IMHO, any such statistics can only be considered significant if it can be shown that the tendency of a decimal-base-1 (9) being followed by a decimal-base+1 (1) is not present in other base systems than decimal In Particular bases which are multiples of small primes such as 2 and 5, say 6,30,14,..
My hunch is that it is.

So consecutive primes with the same last digit is less common than other combinations apparently, so long sequences of primes with the same last digit is like rare gems.

Here is the first occurrence of "n" consecutive primes with the same last digits up to n=14:

[CODE]
[U]Last digit 1:[/U]
n=2: 181-191
n=3: 4831-4871
n=4: 22501-22541
n=5: 216401-216481
n=6: 2229971-2230061
n=7: 3873011-3873151
n=8,9: 36539311-36539501
n=10: 196943081-196943291
n=11: 14293856441-14293856701
n=12,13,14: 154351758091-154351758551

[U]Last digit 3:[/U]
n=2: 283-293
n=3: 6793-6823
n=4: 22963-23003
n=5: 752023-752093
n=6: 2707163-2707283
n=7,8: 44923183-44923313
n=9: 961129823-961129993
n=10: 1147752443-1147752743
n=11: 6879806623-6879806933
n=12: 131145172583-131145172913
n=13: 177746482483-177746482853
n=14: 795537219143-795537219443

[U]Last digit 7:[/U]
n=2: 337-347
n=3: 1627-1657
n=4: 57427-57487
n=5: 192637-192737
n=6: 776257-776357
n=7: 15328637-15328757
n=8: 70275277-70275427
n=9: 244650317-244650617
n=10,11: 452942827-452943157
n=12: 73712513057-73712513627
n=13: 319931193737-319931194127
n=14: 2618698284817-2618698285337

[U]Last digit 9:[/U]
n=2: 139-149
n=3: 3089-3119
n=4: 18839-18899
n=5: 123229-123289
n=6: 2134519-2134609
n=7: 12130109-12130319
n=8: 23884639-23884799
n=9: 363289219-363289379
n=10: 9568590299-9568590529
n=11: 24037796539-24037796789
n=12: 130426565719-130426566079
n=13: 405033487139-405033487499
n=14: 3553144754209-3553144754689
[/CODE]

Terry Tao discussed the paper:
[URL="https://terrytao.wordpress.com/tag/kannan-soundararajan/"]Biases between consecutive primes[/URL]
[QUOTE]However, Lemke Oliver and Soundararajan argue (backed by both plausible heuristic arguments (based ultimately on the Hardy-Littlewood prime tuples conjecture), as well as substantial numerical evidence) that there is a significant bias away from the tuples (1 mod 3, 1 mod 3) and (2 mod 3, 2 mod 3) – informally, adjacent primes don’t like being in the same residue class! [/QUOTE]

I can't decide between 1 or 9 and 3 or 7. How does 1 and 9 remain in the top lead rather than 3 or 7? When does this change? :smile:

[url]http://korn19.ch/coding/primes/ending.php[/url]

John Baez [URL="https://plus.google.com/117663015413546257905/posts/B1w55N6pidw"]commented[/URL]:
[QUOTE]Whoa! The primes are acting weird!

What percent of primes end in a 7? I mean when you write them out in base ten.

Well, if you look at the first hundred million primes, the answer is 25.000401%. That looks suspiciously close to 1/4. And that makes sense, because there are just 4 digits that a prime can end in, unless it's really small: 1, 3, 7 and 9.

So, you might think the endings of prime numbers are random, or very close to it. But 3 days ago two mathematicians shocked the world with a paper that asked some other questions, like this:

[B]If you have a prime that ends in a 7, what's the probability that the next prime ends in a 7?[/B]

I would still expect the answer to be close to 25%. But these mathematicians, Robert Oliver and Kannan Soundarajan, actually looked!

And they found that among the first hundred million primes, the answer is just 17.757%. That's way off!

So if a prime ends in a 7, it seems to somehow tell the next prime "[I]I rather you wouldn't end in a 7. I just did that.[/I]"
[/QUOTE]
[QUOTE]And if you want to go even deeper without reading the actual paper, Terry Tao has a blog article:

[url]https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/[/url]

It starts out readable, and eventually requires extreme persistence and/or skill in number theory.
[/QUOTE]
One interesting thing is that this bias is very obvious once it is pointed out.
So one wonders why it took so long to be noticed.

From the paper:
[QUOTE]Despite the lack of understanding of π (x; q, [B]a[/B]), any model based on the randomness of the primes would suggest strongly that every permissible pattern of [I]r[/I] consecutive primes appears roughly equally often: that is, if [B]a[/B] is an [I]r[/I]-tuple of reduced residue classes (mod q), then π (x; q, [B]a[/B])∼π(x)/φ(q)[SUP]r[/SUP]. However, a look at the data might shake that belief! For example, among the first million primes (for convenience restricting to those greater than 3) we find π(x[SUB]0[/SUB]; 3, (1,1)) = 215,873, π(x[SUB]0[/SUB]; 3, (1,2)) = 283,957, π(x[SUB]0[/SUB]; 3, (2,1)) = 283,957, π(x[SUB]0[/SUB]; 3, (2,2)) = 216,213.

These numbers show substantial deviations from the expectation that all four quantities should be roughly 250,000. Further, Chebyshev’s bias (mod 3) might have suggested a slight preference for the pattern (2,2) over the other possibilities, and this is clearly not the case. The discrepancy observed above persists for larger x, and also exists for other moduli q.

For example, among the first hundred million primes modulo 10, there is substantial deviation from the prediction that each of the 16 pairs (a,b) should have about 6.25 million occurrences. Specifically, with π(x[SUB]0[/SUB]) = 10[SUP]8[/SUP], we find the following.
[CODE]a b | π(x[SUB]0[/SUB];10,(a,b))
1 1 4,623,042
3 7,429,438
7 7,504,612
9 5,442,345
3 1 6,010,982
3 4,442,562
7 7,043,695
9 7,502,896
7 1 6,373,981
3 6,755,195
7 4,439,355
9 7,431,870
9 1 7,991,431
3 6,372,941
7 6,012,739
9 4,622,916[/CODE]
Apart from the fact that the entries vary dramatically (much more than in Chebyshev’s bias), the key feature to be observed in this data is that the diagonal classes (a,a) occur significantly less often than the non-diagonal classes. Chebyshev’s bias (mod 10) states that the residue classes 3 and 7 (mod10) very often contain slightly more primes than the residue classes 1and 9 (mod 10), but curiously in our data the patterns (3,3) and (7,7) appear less frequently than (1,1) and (9,9); this suggests again that a different phenomenon is at play here. The purpose of this paper is to develop a heuristic, based on the Hardy-Littlewood prime k-tuples conjecture, which explains the biases seen above.

We are led to conjecture that, while the primes counted by π(x; q, [B]a[/B]) do have density1/φ(q)[SUP]r[/SUP] in the limit, there are large secondary terms in the asymptotic formula that create biases toward and against certain patterns. The dominant factor in this bias is determined by the number of [I]i[/I] for which a[I][SUB]i+1[/SUB][/I]≡a[SUB][I]i[/I][/SUB] (mod q), but there are also lower order terms that do not have an easy description.[/QUOTE]