20160314, 06:26  #1 
Jun 2003
2^{2}·3·5·7·13 Posts 
Unexpected biases in the distribution of consecutive primes

20160314, 08:15  #2 
"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. 
20160314, 08:31  #3  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
1C35_{16} Posts 
Quote:


20160314, 08:41  #4 
"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 20160314 at 08:49 
20160314, 08:49  #5 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 

20160314, 20:15  #6  
"Jason Goatcher"
Mar 2005
3507_{10} Posts 
Quote:
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 20160314 at 20:17 

20160315, 03:11  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}×3^{3}×11 Posts 
IMHO, any such statistics can only be considered significant if it can be shown that the tendency of a decimalbase1 (9) being followed by a decimalbase+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 20160315 at 03:15 
20160315, 10:54  #8 
Einyen
Dec 2003
Denmark
2·3·5^{2}·23 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: 181191 n=3: 48314871 n=4: 2250122541 n=5: 216401216481 n=6: 22299712230061 n=7: 38730113873151 n=8,9: 3653931136539501 n=10: 196943081196943291 n=11: 1429385644114293856701 n=12,13,14: 154351758091154351758551 Last digit 3: n=2: 283293 n=3: 67936823 n=4: 2296323003 n=5: 752023752093 n=6: 27071632707283 n=7,8: 4492318344923313 n=9: 961129823961129993 n=10: 11477524431147752743 n=11: 68798066236879806933 n=12: 131145172583131145172913 n=13: 177746482483177746482853 n=14: 795537219143795537219443 Last digit 7: n=2: 337347 n=3: 16271657 n=4: 5742757487 n=5: 192637192737 n=6: 776257776357 n=7: 1532863715328757 n=8: 7027527770275427 n=9: 244650317244650617 n=10,11: 452942827452943157 n=12: 7371251305773712513627 n=13: 319931193737319931194127 n=14: 26186982848172618698285337 Last digit 9: n=2: 139149 n=3: 30893119 n=4: 1883918899 n=5: 123229123289 n=6: 21345192134609 n=7: 1213010912130319 n=8: 2388463923884799 n=9: 363289219363289379 n=10: 95685902999568590529 n=11: 2403779653924037796789 n=12: 130426565719130426566079 n=13: 405033487139405033487499 n=14: 35531447542093553144754689 Last fiddled with by ATH on 20160315 at 10:55 
20160315, 20:22  #9  
"Gang aft agley"
Sep 2002
3754_{10} Posts 
Terry Tao discussed the paper:
Biases between consecutive primes Quote:


20160315, 21:09  #10 
"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 
20160316, 08:22  #11  
"Gang aft agley"
Sep 2002
2·1,877 Posts 
John Baez commented:
Quote:
Quote:
So one wonders why it took so long to be noticed. From the paper: Quote:
Last fiddled with by only_human on 20160316 at 08:41 Reason: fussing with my poor tablet skills to try to represent the pdf contents. expanded Baez quote 

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