20230904, 09:09  #1 
Sep 2023
2 Posts 
Dumb conjectures most likely previously discovered
I feel stupid writing this. I am sure I am regurgitating knowledge already known, and there are no doubt conjectures and proofs that already discuss this, but I thought it was interesting and I am going to post it here for other people to have some fun with.
If you calculate the distance between any two consecutive primes as p(n+1)p(n)+1, then the result will appear precisely once as a factor of a composite number between the two primes. The distance may be prime or it may be composite. If composite, the factors of the composite number will appear as the set of factors adjacent to each other within one of the composites between the two primes. For instance: Code:
2 2 dist=2 sqrt=1 factors=0 uniquefactors=0 distoffset=0 distfactors=2 3 3 dist=2 sqrt=1 factors=0 uniquefactors=0 distoffset=0 distfactors=2 4 2,2 5 5 dist=3 sqrt=2 factors=2 uniquefactors=1 distoffset=0 distfactors=3 6 2,3 7 7 dist=3 sqrt=2 factors=2 uniquefactors=2 distoffset=2 distfactors=3 8 2,2,2 9 3,3 10 2,5 11 11 dist=5 sqrt=3 factors=7 uniquefactors=3 distoffset=7 distfactors=5 12 2,2,3 13 13 dist=3 sqrt=3 factors=3 uniquefactors=2 distoffset=3 distfactors=3 14 2,7 15 3,5 16 2,2,2,2 17 17 dist=5 sqrt=4 factors=8 uniquefactors=4 distoffset=4 distfactors=5 18 2,3,3 19 19 dist=3 sqrt=4 factors=3 uniquefactors=2 distoffset=2 distfactors=3 20 2,2,5 21 3,7 22 2,11 23 23 dist=5 sqrt=4 factors=7 uniquefactors=5 distoffset=3 distfactors=5 24 2,2,2,3 25 5,5 26 2,13 27 3,3,3 28 2,2,7 29 29 dist=7 sqrt=5 factors=14 uniquefactors=5 distoffset=14 distfactors=7 30 2,3,5 31 31 dist=3 sqrt=5 factors=3 uniquefactors=3 distoffset=2 distfactors=3 32 2,2,2,2,2 33 3,11 34 2,17 35 5,7 36 2,2,3,3 37 37 dist=7 sqrt=6 factors=15 uniquefactors=6 distoffset=11 distfactors=7 38 2,19 39 3,13 40 2,2,2,5 41 41 dist=5 sqrt=6 factors=8 uniquefactors=5 distoffset=8 distfactors=5 42 2,3,7 43 43 dist=3 sqrt=6 factors=3 uniquefactors=3 distoffset=2 distfactors=3 44 2,2,11 45 3,3,5 46 2,23 47 47 dist=5 sqrt=6 factors=8 uniquefactors=5 distoffset=6 distfactors=5 48 2,2,2,2,3 49 7,7 50 2,5,5 51 3,17 52 2,2,13 53 53 dist=7 sqrt=7 factors=15 uniquefactors=6 distoffset=6 distfactors=7 54 2,3,3,3 55 5,11 56 2,2,2,7 57 3,19 58 2,29 59 59 dist=7 sqrt=7 factors=14 uniquefactors=7 distoffset=10 distfactors=7 60 2,2,3,5 61 61 dist=3 sqrt=7 factors=4 uniquefactors=3 distoffset=3 distfactors=3 We can also see that distances are predominantly prime. We can also state precisely when a distance that is prime can start showing up, but not precisely when it will show up. Sometimes a distance that is prime that is of a higher value will show up before it, e.g. a distance of 19 will show up at prime #541, but a distance of 17 will show up prime #1847. But we can categorically state that a distance of 19 won't show up until it is permitted too. We can see that distances within a wheel are palindromic. If we calculate a wheel of 30 from the first 3 primes, 2,3,5, we have a repeating pattern that is 30 digits (the product of 2,3,5) in length. If we lay it out like a bit pattern. Code:
1 2 3 4 5 6 123456789012345678901234567890123456789012345678901234567890123456785 2 101010101010101010101010101010101010101010101010101010101010101010101 3 110110110110110110110110110110110110110110110110110110110110110110110 5 11110111101111X111101111011110111101111011110111101111011110111101110 If we take the sequence of bits for the wheel of 30, we see not only is it a palindrome, but that there are repeating subphrases, and in fact, these subphrases repeat predictably throughout the number line. Similarly, if we treat this bit pattern in a particular way, which is modified Pritchard's algorithm, we calculate the jumps between primes. Not only is it possible to calculate the jumps between primes, but we can calculate the next jumps between primes: Code:
jumps: 3 2 [2, 4] jumps: 5 6 [ 4, 2, 4, 2, 4, 6] jumps: 7 11 [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 10] jumps: 11 25 [ 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 12] jumps: 13 33 [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 16] jumps: 17 54 [ 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 18] jumps: 19 64 [ 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 22] jumps: 23 90 [ 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 28] jumps: 29 136 [ 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 30] jumps: 31 151 [ 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 36] jumps: 37 207 [ 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 40] jumps: 41 250 [ 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 6, 8, 18, 10, 14, 4, 2, 4, 6, 8, 4, 2, 6, 12, 10, 2, 4, 2, 4, 6, 12, 12, 8, 12, 6, 4, 6, 8, 4, 8, 4, 14, 4, 6, 2, 4, 6, 2, 6, 10, 20, 6, 4, 2, 42] jumps: 43 269 [ 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 6, 8, 18, 10, 14, 4, 2, 4, 6, 8, 4, 2, 6, 12, 10, 2, 4, 2, 4, 6, 12, 12, 8, 12, 6, 4, 6, 8, 4, 8, 4, 14, 4, 6, 2, 4, 6, 2, 6, 10, 20, 6, 4, 2, 24, 4, 2, 10, 12, 2, 10, 8, 6, 6, 6, 18, 6, 4, 2, 12, 10, 12, 8, 16, 46] jumps: 47 314 [ 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 6, 8, 18, 10, 14, 4, 2, 4, 6, 8, 4, 2, 6, 12, 10, 2, 4, 2, 4, 6, 12, 12, 8, 12, 6, 4, 6, 8, 4, 8, 4, 14, 4, 6, 2, 4, 6, 2, 6, 10, 20, 6, 4, 2, 24, 4, 2, 10, 12, 2, 10, 8, 6, 6, 6, 18, 6, 4, 2, 12, 10, 12, 8, 16, 14, 6, 4, 2, 4, 2, 10, 12, 6, 6, 18, 2, 16, 2, 22, 6, 8, 6, 4, 2, 4, 8, 6, 10, 2, 10, 14, 10, 6, 12, 2, 4, 2, 10, 12, 2, 16, 2, 6, 4, 2, 10, 8, 18, 24, 4, 52] Going back to the idea that prime gaps encode the size of the gap, and that the size of the gap can only appear once between two primes, we can take any composite number with known factors, adjust a single prime by one step, e.g. 2,3,5 becomes 3,3,5, and then state that one or more primes lay between those two composite numbers. Unrelated to the above, but there's an interesting Wile E. Coyote type of algorithm that if you have only the product of all previous primes (which quickly becomes a stupendously large number), and the last prime you found, you can trivially predict the number prime in sequence with only a few simple operations. 
20230904, 11:53  #2  
Feb 2017
Nowhere
23·283 Posts 
Quote:
It appears that  in the enduring tradition of "I'm a number theorist, I can't count"  the index of the "distance" in your table is offset from your formula by 1. No matter  your observation is correct, for a very simple reason. Your distance function d = p_{n+1}  p_{n} + 1 (which is 1 more than the length of the gap) is equal to the number of consecutive positive integers in the list p_{n}, p_{n} + 1, ..., p_{n+1}. For a given positive integer d, exactly one of any d consecutive positive integers is a multiple of d. In your case, it can't be either endpoint. And therefore, the multiple of d is somewhere in between the endpoints. 

20230904, 16:47  #3 
Sep 2023
2 Posts 
Yep, I regret posting and trying to engage in a conversation. If only there were a delete post button.

20230904, 18:15  #4 
"Curtis"
Feb 2005
Riverside, CA
53×113 Posts 
Posting with modesty, as you have, is nothing to be ashamed of nor to feel "dumb" for.
I think it's cool that you took the approach of "I found a pattern, could someone tell me how it relates to known math?" 
20230905, 12:30  #5  
Feb 2017
Nowhere
23·283 Posts 
Quote:
You have no reason to regret your post. Your observation was, after all, correct. And we all (myself included) have overlooked simple explanations of our observations or guesses that are "obvious" (at least in retrospect). I have lost count of the number of times I have felt compelled to acknowledge such explanations of my own observations with responses featuring "D'oh!" Sometimes, that's how we learn. So keep your observations coming. NOTE: I have also lost count of the number of posters who offer nonsensical ideas, and insist they are right even after having been proven to be wrong. Their offerings often wind up in the "Miscellaneous Math" area. Sometimes they are exiled to "Blogorrhea." Occasionally, they have seen the light (or felt the heat) and become  deservedly  ashamed or embarrassed. Sometimes, they request the history of their foolishness to be excised. Such requests are always denied. 

20230909, 23:59  #6 
May 2018
100110010_{2} Posts 

20230910, 00:02  #7 
"Forget I exist"
Jul 2009
Dartmouth NS
5·13·131 Posts 
There is. It only shows if you go to edit though. Which is possible any time you are last to post on a thread. You could flag as spam ( exclamation mark in yield sign to left side of post ...)
Last fiddled with by science_man_88 on 20230910 at 00:26 
20230910, 03:02  #8  
"Curtis"
Feb 2005
Riverside, CA
53×113 Posts 
Quote:
Marking a post spam sends an alert to *every* mod. That's not a good plan for one's own post, particularly since mods have made clear that this forum prefers to leave posts undeleted. If you don't like that, think before you post. 

20230910, 07:19  #9 
"Composite as Heck"
Oct 2017
2^{4}·61 Posts 
Posting might peg you as having worse understanding than others but who cares, it's not a competition. As long as you want to learn the more you post the more your poor ideas get debunked and you get nudged in the right direction. Also posting should force you to think about the idea from another's perspective to try and convey the idea in a way they can understand, so you start to learn the language. That's why I don't mind posting openly my crappy code from time to time, even if no one looks at it it gives me motivation to think critically and improve.

20230910, 13:18  #10  
Feb 2017
Nowhere
6509_{10} Posts 
Quote:
There are two obvious reasons that members (without extra privileges) aren't allowed to edit or delete their own posts later on. One is, we want to encourage members to think before they post. One way we do this is restricting editing to within an hour after posting. If you discover later on that you've posted something that has a faulty argument, states a false result, or merely overlooks a simple explanation, you aren't allowed to delete the post simply because you're embarrassed at your mistake. You'll have to live with your embarrassment. If you're willing to learn from your mistake, nobody will hold it against you. Another reason is (no reflection on you of course!), there have been those who who would use the ability to edit or delete posts to rewrite history. If they posted a faulty argument and/or wrong result, they would simply delete their post, and then turn around and claim, "I never said that." Or they might rewrite their post after the fact so as to destroy the evidence. This would not, of course, rewrite portions of the original post already quoted in responses, but it would cause the quotations not to match the rewritten post. 

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