144 (number)

144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145.

It represents a dozen dozens, or one gross. It is the number of square inches in a square foot.

In number theory, 144 is the twelfth Fibonacci number; it is the only Fibonacci number (other than 0, and 1) to also be a square.

Mathematics
144 is the square of 12. It is also the twelfth Fibonacci number, following 89 and preceding 233, and the only Fibonacci number (other than 0, and 1) to also be a square. 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 has 16 divisors. 144 is also equal to the sum of the eighth twin prime pair, (71 + 73). It is divisible by the value of its φ function, which returns 48 in its case, and there are 21 solutions to the equation $$\varphi (x) = 144.$$ This is more than any integer below it, which makes it a highly totient number. In decimal, 144 is the largest of only four sum-product numbers, and it is a Harshad number, where $$1 + 4 + 4 = 9$$, which divides 144.

Powers
144 is the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture. It was famously published in a paper by both authors, whose body consisted of only two sentences:

"L. J. Lander and T. R. Parkin, p. 1079"

In decimal notation, when each of the digits in the expression of the square of twelve are reversed, the equation remains true:


 * $$144 = 12 \times 12$$
 * $$441 = 21 \times 21$$

Another number that shares this property is 169, where $$13 \times 13 = 169$$, while $$31 \times 31 = 961.$$

Geometry
A regular ten-sided decagon has an internal angle of 144 degrees, which is equal to four times its own central angle, and equivalently twice the central angle of a regular five-sided pentagon, while in four dimensions, the snub 24-cell, one of three semiregular polytopes in the fourth dimension, contains a total of 144 polyhedral cells: 120 regular tetrahedra and 24 regular icosahedra. Meanwhile, the maximum determinant in a 9 by 9 matrix of zeroes and ones is 144.

In the Leech lattice
144 is the sum of the divisors of 70: $$\sigma (70) = 144$$, where 70 is part of the only solution to the cannonball problem aside from the trivial solution, in-which the sum of the squares of the first twenty-four integers is equal to the square of another integer, 70 — and meaningful in the context of constructing the Leech lattice in twenty-four dimensions via the Lorentzian even unimodular lattice II25,1. 144 is relevant in testing whether two vectors in the quaternionic Leech lattice are equivalent under its automorphism group, Conway group $$Co_{0}$$: modulo $$1 + i$$, every vector is congruent to either $$0$$ or a minimal vector that is one of $$196,560 \div {144} = 1,365$$ algebraic coordinate-frames, in-which a frame sought can be carried to its standard frame that is then checked for equivalence under a group stabilizing the frame of interest.

Other fields



 * Sonnet 144 by William Shakespeare.
 * 1:144 scale is a scale used for some scale models.
 * Mahjong is usually played with a set of 144 tiles.
 * The measurement, in cubits, of the wall of New Jerusalem shown by the seventh angel (Bible, Revelation 21:17). 144 also occurs in the name of Psalm 144.