# 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:

 $\begin{array}[]{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\ &&&&&&&&1&&2&&&&&&&\\ &&&&&&&1&&2&&3&&&&&&\\ &&&&&&1&&2&&3&&4&&&&&\\ &&&&&1&&4&&3&&2&&5&&&&\\ &&&&1&&4&&3&&2&&5&&6&&&\\ &&&1&&4&&3&&2&&5&&6&&7&&\\ &&1&&4&&7&&6&&5&&2&&3&&8&\\ &&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\ \end{array}$

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 PrimePyramid 2013-03-22 17:00:03 2013-03-22 17:00:03 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A41