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| ||||
|---|---|---|---|---|
| Cardinal | one hundred forty-four | |||
| Ordinal | 144th (one hundred forty-fourth) | |||
| Factorization | 24 × 32 | |||
| Divisors | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 | |||
| Greek numeral | ΡΜΔ´ | |||
| Roman numeral | CXLIV | |||
| Binary | 100100002 | |||
| Ternary | 121003 | |||
| Senary | 4006 | |||
| Octal | 2208 | |||
| Duodecimal | 10012 | |||
| Hexadecimal | 9016 | |||
144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145. It is a dozen dozens, or one gross.
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.[1][2] 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 contains 16.[3] 144 is also equal to the sum of the eighth twin prime pair, (71 + 73).[4][5] It is divisible by the value of its φ function, which returns 48 in its case,[6] and there are 21 solutions to the equation φ(x) = 144. This is more than any integer below it, which makes it a highly totient number.[7]
As a square number in decimal notation, 144 = 12 × 12, and if each number is reversed the equation still holds: 21 × 21 = 441. 169 shares this property, 13 × 13 = 169, while 31 × 31 = 961. Also in decimal, 144 is the largest of only four sum-product numbers,[8] and it is a Harshad number, since 1 + 4 + 4 = 9, which divides 144.[9]
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:[10]
A direct search on the CDC 6600 yielded
275 + 845 + 105 + 1335 = 1445
as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler that at least n nth powers are required to sum to an nth power, n > 2.
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.
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.
The maximum determinant in a 9 by 9 matrix of zeroes and ones is 144.[11]
In particular, 144 is the sum of the divisors of 70: σ(70) = 144,[12] 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.[13] Furthermore, 144 is relevant in testing whether two vectors in the quaternionic Leech lattice are equivalent under its automorphism group, Conway group Co0: modulo 1 + i, every vector is congruent to either 0 or a minimal vector that is one of 196,560 ÷ 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.[14][15][16]
144 is also: