As you know, every integer greater than \(1\) can be uniquely factored into a product of powers of prime numbers: \(24 = 8 \times 3 = 2^3 \times 3^1\), that kind of thing. No news there. Knowing that each prime number can be indexed, as the first prime \(p_1 = 2\), the second prime \(p_2 = 3\), the third prime \(p_3 = 5\), the fourth prime \(p_4 = 7\), and so on, each integer greater than \(1\) can then be expressed as

$$\prod p_i^k$$

where \(i\) is the index of the prime and \(k\) is the exponent of the prime in the integer factorization. For example, \(24 = 8 \times 3 = 2^3 \times 3^1 = p_1^3 p_2^1\). No news there either.

What you might not have tried before is to then do the same thing to the indexes and exponents: \(24 = 8 \times 3 = 2^3 \times 3^1 = p_1^3 p_2^1 = p_1^{p_2^1} p_{p_1^1}^1\).

If you keep going, eventually all of the indexes and exponents will be \(1\)’s: \(24 = 8 \times 3 = 2^3 \times 3^1 = p_1^3 p_2^1 = p_1^{p_2^1} p_{p_1^1}^1 = p_1^{p_{p_1^1}^1} p_{p_1^1}^1\).

Once you get that far, you might as well remove all of the 1’s because they don’t add any information, leaving you with \(p_1^{p_{p_1^1}^1} p_{p_1^1}^1 = p^{p_p} p_p\). Using this procedure, any integer greater than 1 can be expressed using this kind of expression.

\(2\) | \(= p_1 = p\) |

\(3\) | \(= p_2 = p_p\) |

\(4\) | \(= p_1^2 = p_1^{p_1} = p^p\) |

\(5\) | \(= p_3 = p_{p_2} = p_{p_p}\) |

\(6\) | \(= p_1 p_2 = p p_p\) |

\(7\) | \(= p_4 = p_{p^p}\) |

\(8\) | \(= p_1^3 = p^{p_2} = p^{p_p}\) |

\(9\) | \(= p_2^2 = p_p^p\) |

\(10\) | \(= p_1 p_3 = p p_{p_p}\) |

\(11\) | \(= p_5 = p_{p_{p_p}}\) |

Focusing on the shape of the expressions, towers of exponents look like this (just towers of powers of 2):

\(p\) | \(= 2\) |

\(p^p\) | \(= 2^2 = 4\) |

\(p^{p^p}\) | \(= 2^{2^2} = 16\) |

\(p^{p^{p^p}}\) | \(= 2^{2^{2^2}} = 65,536\) |

\(p^{p^{p^{p^p}}}\) | \(= 2^{2^{2^{2^2}}} \approx 2.00352993 \times 10^{19,728}\) |

And towers of indexes/subscripts look like this:

\(p\) | \(= 2\) |

\(p_p\) | \(= 3\) |

\(p_{p_p}\) | \(= 5\) |

\(p_{p_{p_p}}\) | \(= 11\) |

\(p_{p_{p_{p_p}}}\) | \(= 31\) |

\(p_{p_{p_{p_{p_p}}}}\) | \(= 127\) |

\(p_{p_{p_{p_{p_{p_p}}}}}\) | \(= 709\) |

\(p_{p_{p_{p_{p_{p_{p_p}}}}}}\) | \(= 5,381\) |

\(p_{p_{p_{p_{p_{p_{p_{p_p}}}}}}}\) | \(= 52,711\) |

\(p_{p_{p_{p_{p_{p_{p_{p_{p_p}}}}}}}}\) | \(= 648,391\) |

\(p_{p_{p_{p_{p_{p_{p_{p_{p_{p_p}}}}}}}}}\) | \(= 9,737,333\) |

This is OEIS sequence A007097,
\(a(n+1)\) is the \(a(n)\)^{th} prime number.

Combined indexes and exponents:

\(p_p^p\) | \(= p_2^2 = 3^2 = 9\) |

\(p_{p_p}^{p^p}\) | \(= p_3^{2^2} = 5^4 = 625\) |

\(p_{p_{p_p}}^{p^{p^p}}\) | \(= \ldots\) |

Some of these expressions look like typeset versions of the Sierpinski gasket:

\(p\) | \(= 2\) |

\(p_p^p\) | \(= 9\) |

\(p_{p_p^p}^{p_p^p}\) | \(= 1,801,152,661,463\) |

\(p_{p_{p_p^p}^{p_p^p}}^{p_{p_p^p}^{p_p^p}}\) | \(= {55,125,235,480,573}^{1,801,152,661,463} \approx 10^{10^{13.39\ldots}}\) |

Things might have got a little bit out of hand.

I didn’t really add anything to the work of J. Awbrey, Riffs and Rotes, but still.

Designed and rendered using Wolfram Mathematica 11, with first timer’s thanks to:

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First draft August 2018 by Robert Dickau.

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