prime pyramid
A prime pyramid is a triangular arrangement of numbers in which each row $n$ has the integers from 1 to $n$ but in an order such that the sum of any two consecutive terms in a row is a prime number^{}. The first number must be 1, and the last number is usually required to be $n$, all the numbers in between are in whatever order fulfills the requirement for prime sums. Unlike other triangular arrangements of numbers like Pascal’s triangle or Losanitsch’s triangle, the contents of a given row are not determined by those of the previous row. However, if it happens that one has calculated row $n-1$ and that $2n-1$ is a prime number, one could just copy the previous row and add $n$ at the end. Here is a prime pyramid reckoned that way:
$$\begin{array}{cccccccccccccccccc}& & & & & & & & & \hfill 1\hfill & & & & & & & & \\ & & & & & & & & \hfill 1\hfill & & \hfill 2\hfill & & & & & & & \\ & & & & & & & \hfill 1\hfill & & \hfill 2\hfill & & \hfill 3\hfill & & & & & & \\ & & & & & & \hfill 1\hfill & & \hfill 2\hfill & & \hfill 3\hfill & & \hfill 4\hfill & & & & & \\ & & & & & \hfill 1\hfill & & \hfill 4\hfill & & \hfill 3\hfill & & \hfill 2\hfill & & \hfill 5\hfill & & & & \\ & & & & \hfill 1\hfill & & \hfill 4\hfill & & \hfill 3\hfill & & \hfill 2\hfill & & \hfill 5\hfill & & \hfill 6\hfill & & & \\ & & & \hfill 1\hfill & & \hfill 4\hfill & & \hfill 3\hfill & & \hfill 2\hfill & & \hfill 5\hfill & & \hfill 6\hfill & & \hfill 7\hfill & & \\ & & \hfill 1\hfill & & \hfill 4\hfill & & \hfill 7\hfill & & \hfill 6\hfill & & \hfill 5\hfill & & \hfill 2\hfill & & \hfill 3\hfill & & \hfill 8\hfill & \\ & & & & & \hfill \mathrm{\vdots}\hfill & & & & \hfill \mathrm{\vdots}\hfill & & & & \hfill \mathrm{\vdots}\hfill & & & & \end{array}$$ |
Often row 1 just contains an asterisk or some other non-numerical symbol, but since the idea of adding two numbers in row 1 is moot, here row 1 just contains a 1 per analogy to the following rows and to other triangular arrangements of numbers.
References
- 1 R. K. Guy, Unsolved Problems in Number Theory^{} New York: Springer-Verlag 2004: C1
Title | prime pyramid |
---|---|
Canonical name | PrimePyramid |
Date of creation | 2013-03-22 17:00:03 |
Last modified on | 2013-03-22 17:00:03 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 4 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 11A41 |