mersenneforum.org Unexpected biases in the distribution of consecutive primes
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 2016-03-14, 06:26 #1 axn     Jun 2003 22×32×151 Posts Unexpected biases in the distribution of consecutive primes
 2016-03-14, 08:15 #2 PawnProver44     "NOT A TROLL" Mar 2016 California 197 Posts 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.
2016-03-14, 08:31   #3
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3×29×83 Posts

Quote:
 Originally Posted by PawnProver44 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.
That's like saying "based on this link that says A is true, I therefore think this is most likely true: A".

 2016-03-14, 08:41 #4 PawnProver44     "NOT A TROLL" Mar 2016 California 197 Posts 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.) Last fiddled with by PawnProver44 on 2016-03-14 at 08:49
2016-03-14, 08:49   #5
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·29·83 Posts

Quote:
 Originally Posted by PawnProver44 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.
Did you read the paper or the article?

2016-03-14, 20:15   #6
jasong

"Jason Goatcher"
Mar 2005

66638 Posts

Quote:
 Originally Posted by axn
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. :)

Last fiddled with by jasong on 2016-03-14 at 20:17

 2016-03-15, 03:11 #7 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 2,333 Posts 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. Last fiddled with by a1call on 2016-03-15 at 03:15
 2016-03-15, 10:54 #8 ATH Einyen     Dec 2003 Denmark 2·17·101 Posts 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: Last digit 1: 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 Last digit 3: 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 Last digit 7: 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 Last digit 9: 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 Last fiddled with by ATH on 2016-03-15 at 10:55
2016-03-15, 20:22   #9
only_human

"Gang aft agley"
Sep 2002

2×1,877 Posts

Terry Tao discussed the paper:
Biases between consecutive primes
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!

 2016-03-15, 21:09 #10 PawnProver44     "NOT A TROLL" Mar 2016 California 197 Posts 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? http://korn19.ch/coding/primes/ending.php
2016-03-16, 08:22   #11
only_human

"Gang aft agley"
Sep 2002

1110101010102 Posts

John Baez commented:
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: If you have a prime that ends in a 7, what's the probability that the next prime ends in a 7? 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 rather you wouldn't end in a 7. I just did that."
Quote:
 And if you want to go even deeper without reading the actual paper, Terry Tao has a blog article: https://terrytao.wordpress.com/2016/...cutive-primes/ It starts out readable, and eventually requires extreme persistence and/or skill in number theory.﻿
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, a), any model based on the randomness of the primes would suggest strongly that every permissible pattern of r consecutive primes appears roughly equally often: that is, if a is an r-tuple of reduced residue classes (mod q), then π (x; q, a)∼π(x)/φ(q)r. 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(x0; 3, (1,1)) = 215,873, π(x0; 3, (1,2)) = 283,957, π(x0; 3, (2,1)) = 283,957, π(x0; 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(x0) = 108, we ﬁnd the following. Code: a b | π(x0;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 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, a) do have density1/φ(q)r 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 for which ai+1≡a[I]i[/I] (mod q), but there are also lower order terms that do not have an easy description.

Last fiddled with by only_human on 2016-03-16 at 08:41 Reason: fussing with my poor tablet skills to try to represent the pdf contents. expanded Baez quote

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