6 ms·

Semi-related, I once noticed an odd thing about low prime numbers which (as far as I know) nobody else has made note of: if you look at the natural numbers 1, 2

Semi-related, I once noticed an odd thing about low prime numbers which (as far as I know) nobody else has made note of: if you look at the natural numbers 1, 2, 3,...n, and keep track of how far each one is from a prime number, there are a lot of palindromic sequences in the progression. https://math.stackexchange.com/questions/1338059/why-are-there-palindromic-subsequences-at-random-among-this-sequence https://math.stackexchange.com/questions/1338059/why-are-the... I am not particularly talented with programming mathematical quandaries, but if anyone else wanted to analyze this further, it could perhaps be an avenue to interesting results.

soVeryTired 8y ago
I think your obervation is due to palindromes in the differences between primes themselves. I worked through your example for the integers 8 to 16, the sequence being 3 2 3 0 1 0 3 2 3. The sequence of numbers we observe is due entirely to the locations of the primes 5, 7, 11, 13, 17, and 19. The differences between these primes is 2, 4, 2, 4, 2, which is again a palindrome. I wouldn't be surprised if that's the reason your original sequence is palindromic. If it is, the question is then whether palindromes in distances between primes are unusually common.
- aerovistae 8y ago
  You are correct! Great observation.
  - soVeryTired 8y ago
    Weird, someone else pointed out the same thing on stackexchange around the same time I did. I guess they must have come from here.
    - aerovistae 8y ago
      I was wondering if you were the same person with different account names on different sites! Evidently not.
    - DavidSJ 8y ago
      I also came close to pointing that out, and then I noticed that it only makes palindromes of this sequence more probable, rather than guaranteeing them (see my comment on Math SE).
andrewla 8y ago
It looks like the Online Encyclopedia of Integer Sequences has some information on your series [1]. Always an interesting place to start going down a rabbit hole. [1] https://oeis.org/A047160 https://oeis.org/A047160
- daveguy 8y ago
  And this one with just the distances between primes: https://oeis.org/A001223 https://oeis.org/A001223 A lot of references for that sequence. Fascinating stuff... The palindrome observation of the OP might just be that we are good at finding patterns and there are limited numbers of distances with lower numbered primes (and from the graph of your reference it looks like a lot of the distances are concentrated in smaller numbers -- max dist increasing logarithmically?) I wonder if the palindrome conjecture can be posed mathematically or algorithmically (seems like it should). It would be interesting to test palindromes to a certain number compared to palindromes of a similarly distributed set of random numbers. Counting all palindromes of any size will probably be a significant big O. Now I'm just trying to keep myself from looking at the Collatz Conjecture. :)
  - mgsouth 8y ago
    Prime series are palindromic. Given the set of primes (2, 3, 5, ..., p[m]), there is a repeating pattern of length r = 2 * 3 * ... * p[m], where nr +/- [ p[m+1], p[m+2], ... p[m+q] ] (p[m+q] < r/2) are relatively prime to the given set. When n = 1, the only relatively prime integers which aren't truly prime are those which are some combination of two or more p[m+1] ... p[m+q]; the smallest is of course p[m+1] ^ 2.
TheOtherHobbes 8y ago
Possibly related: https://en.wikipedia.org/wiki/Riemann_hypothesis https://en.wikipedia.org/wiki/Riemann_hypothesis

soVeryTired 8y ago

I think your obervation is due to palindromes in the differences between primes themselves. I worked through your example for the integers 8 to 16, the sequence being 3 2 3 0 1 0 3 2 3. The sequence of numbers we observe is due entirely to the locations of the primes 5, 7, 11, 13, 17, and 19. The differences between these primes is 2, 4, 2, 4, 2, which is again a palindrome. I wouldn't be surprised if that's the reason your original sequence is palindromic. If it is, the question is then whether palindromes in distances between primes are unusually common.

aerovistae 8y ago
You are correct! Great observation.
- soVeryTired 8y ago
  Weird, someone else pointed out the same thing on stackexchange around the same time I did. I guess they must have come from here.
  - aerovistae 8y ago
    I was wondering if you were the same person with different account names on different sites! Evidently not.
  - DavidSJ 8y ago
    I also came close to pointing that out, and then I noticed that it only makes palindromes of this sequence more probable, rather than guaranteeing them (see my comment on Math SE).

andrewla 8y ago

It looks like the Online Encyclopedia of Integer Sequences has some information on your series [1]. Always an interesting place to start going down a rabbit hole. [1] https://oeis.org/A047160 https://oeis.org/A047160

daveguy 8y ago
And this one with just the distances between primes: https://oeis.org/A001223 https://oeis.org/A001223 A lot of references for that sequence. Fascinating stuff... The palindrome observation of the OP might just be that we are good at finding patterns and there are limited numbers of distances with lower numbered primes (and from the graph of your reference it looks like a lot of the distances are concentrated in smaller numbers -- max dist increasing logarithmically?) I wonder if the palindrome conjecture can be posed mathematically or algorithmically (seems like it should). It would be interesting to test palindromes to a certain number compared to palindromes of a similarly distributed set of random numbers. Counting all palindromes of any size will probably be a significant big O. Now I'm just trying to keep myself from looking at the Collatz Conjecture. :)
- mgsouth 8y ago
  Prime series are palindromic. Given the set of primes (2, 3, 5, ..., p[m]), there is a repeating pattern of length r = 2 * 3 * ... * p[m], where nr +/- [ p[m+1], p[m+2], ... p[m+q] ] (p[m+q] < r/2) are relatively prime to the given set. When n = 1, the only relatively prime integers which aren't truly prime are those which are some combination of two or more p[m+1] ... p[m+q]; the smallest is of course p[m+1] ^ 2.

TheOtherHobbes 8y ago

Possibly related: https://en.wikipedia.org/wiki/Riemann_hypothesis https://en.wikipedia.org/wiki/Riemann_hypothesis