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Definition

The Burz Prime-Number Pyramid is a triangular arrangement of consecutive integers, structured such that the length of each row corresponds to a prime number, except the first row which contains only the number 1.

Row Structure:

Let p(i) denote the i-th prime number and p(0) = 1.

Prime Distribution in the Pyramid:

Within each row, primes are frequently found in the first, second, last, or second-to-last positions.

Example:

Row 1: p(0) = 1, Row = {1}.

Row 2: p(1) = 2, Row = {2, 3}.

Row 3: p(2) = 3, Row = {4, 5, 6}.

Row 4: p(3) = 5, Row = {7, 8, 9, 10, 11}.

Row 5: p(4) = 7, Row = {12, 13, 14, 15, 16, 17, 18}.

Row 6: p(5) = 11, Row = {19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}.

Row 7: p(6) = 13, Row = {30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42}.

Row 7: p(7) = 17, Row = {43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59}.

Notable Observation

Row extremities seem to contain primes:

First and second positions of each row. (2, 5, 7, 13, 19, 31, 43)

Last and second-to-last positions of each row. (3, 5, 11, 17, 29, 41, 59)

Research Questions

Why are primes concentrated near the edges of each row?

Can this arrangement reveal deeper insights into the distribution of primes?

If you have the time please share your thoughts, it's been something that I had on my mind for years and I would like to read more about this pattern if someone has already analyzed it before. Thank you!

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  • $\begingroup$ This is not a research-level question, sorry. You forgot a lot of primes here (23, 37, 37, 51, 53...) and these do not at all cluster at the extremities. More generally, since row size grows as nlog(n) you will simply see properties of prime distribution in that window. $\endgroup$
    – Jon23
    Commented 6 hours ago
  • $\begingroup$ You are correct, I just find it intriguing that primes apear on the first and last position followed by the second and second-to-last on the next row. I don't think it's a random distribution and I think there might be something more to it. $\endgroup$
    – M B
    Commented 6 hours ago
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    $\begingroup$ One of the first sins when studying primes is to only look at way too little data.There are infinitely many primes, and they are rarefying in your setup (the length of the row grows faster than the number of primes in the row), so it is certain and obvious that the pattern you see will disappear for very large rows, most large rows will be empty of primes... $\endgroup$
    – Jon23
    Commented 5 hours ago
  • $\begingroup$ ... in their first or last 100 positions for instance. But plausibly, I would say, rather than certainly. $\endgroup$
    – Claude Chaunier
    Commented 4 hours ago

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