Dumb conjectures most likely previously discovered

justinlloyd · 2023-09-04, 09:09

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 if we take any two non-consecutive primes, and calculate the distance in the same fashion, the distance appears as a factor precisely once between the two non-consecutive primes.

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

We see that the pattern repeats every 30 bits, but at the but if you take the 15th bit, I've placed an X on the row designated 5, we see the bit pattern goes in reverse. This palindromic pattern holds true out to products of primes with billions upon billions of bits.

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 sub-phrases, and in fact, these sub-phrases 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]

As you can see, we take the first value of the previous sequence, and add it to the value in the end of a sequence, perform the obvious transformation steps that I won't discuss here, and we end up with the new jump to the next prime, and then use those subphrasings discussed earlier, and the palindromic nature, to extend our jumps out. We can predictably list out our jumps to the square of the last prime -1 in the previous sequence before our prediction becomes unreliable. We also need to throw away the first number from the last sequence in the current sequence. It effectively gets consumed.

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.

Dr Sardonicus · 2023-09-04, 11:53

Quote:

Originally Posted by justinlloyd

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.
<snip>
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
<snip>

<snip>

Welcome to the Forum.

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.

justinlloyd · 2023-09-04, 16:47

Yep, I regret posting and trying to engage in a conversation. If only there were a delete post button.

VBCurtis · 2023-09-04, 18:15

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?"

Dr Sardonicus · 2023-09-05, 12:30

Quote:

Originally Posted by justinlloyd

Yep, I regret posting and trying to engage in a conversation. If only there were a delete post button.

What VBCurtis said.

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.

Bobby Jacobs · 2023-09-09, 23:59

Quote:

Originally Posted by justinlloyd

Yep, I regret posting and trying to engage in a conversation. If only there were a delete post button.

I sometimes wish that there was a delete post button, too. Why is there not one?

science_man_88 · 2023-09-10, 00:02

Quote:

Originally Posted by Bobby Jacobs

I sometimes wish that there was a delete post button, too. Why is there not one?

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 ...)

VBCurtis · 2023-09-10, 03:02

Quote:

Originally Posted by science_man_88

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 ...)

This is not true. "edit" is available for one hour after posting.
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.

M344587487 · 2023-09-10, 07:19

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.

Dr Sardonicus · 2023-09-10, 13:18

Quote:

Originally Posted by Bobby Jacobs

I sometimes wish that there was a delete post button, too. Why is there not one?

As already indicated, any poster can edit or delete posts within an hour after posting. This is to allow for correcting typos or mistakenly hitting "submit reply" instead of "preview" immediately after posting.

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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2023-09-04, 16:47	#3
justinlloyd Sep 2023 2 Posts	Yep, I regret posting and trying to engage in a conversation. If only there were a delete post button.

2023-09-04, 18:15	#4
VBCurtis "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?"

2023-09-10, 07:19	#9
M344587487 "Composite as Heck" Oct 2017 2⁴·61 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.