153 (number)

The number 153 is associated with the geometric shape known as the Vesica piscis or Mandorla. Archimedes, in his Measurement of a Circle, referred to this ratio (153/265), as constituting the "measure of the fish", this ratio being an imperfect representation of ${\displaystyle 1/{\sqrt {3}}\approx 0.57735}$.^[2]

As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials: ${\displaystyle 1!+2!+3!+4!+5!}$.^[1]

The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.

The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.

Since ${\displaystyle 153=1^{3}+5^{3}+3^{3}}$, it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits.^[3] Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407.^[4] It is also a Friedman number, since 153 = 3 × 51.

The Biggs–Smith graph is a symmetric graph with 153 edges, all equivalent.

Another feature of the number 153 is that it is the limit of the following algorithm:^[5][6][7]

1. Take a random positive integer, divisible by three
2. Split that number into its base 10 digits
3. Take the sum of their cubes
4. Go back to the second step

An example, starting with the number 84:

${\displaystyle {\begin{aligned}8^{3}+4^{3}&=&512+64&=&576\\5^{3}+7^{3}+6^{3}&=&125+343+216&=&684\\6^{3}+8^{3}+4^{3}&=&216+512+64&=&792\\7^{3}+9^{3}+2^{3}&=&343+729+8&=&1080\\1^{3}+0^{3}+8^{3}+0^{3}&=&1+0+512+0&=&513\\5^{3}+1^{3}+3^{3}&=&125+1+27&=&153\\1^{3}+5^{3}+3^{3}&=&1+125+27&=&153\end{aligned}}}$

There are 153 uniform polypeta that are generated from four different fundamental Coxeter groups in six-dimensional space.

The sum of the first eight Heegner numbers is 153.

The Gospel of John (chapter 21:1–14) includes the miraculous catch of 153 fish as the third appearance of Jesus after his resurrection.^[8] Augustine of Hippo argued that the significance lay in the fact that 153 is the sum of the first 17 integers (i.e. 153 is the 17th triangular number), representing the combination of divine grace (the seven gifts of the Holy Spirit) and law (the Ten Commandments).^[9]