← 143 144 145 →
Cardinalone hundred forty-four
Ordinal144th
(one hundred forty-fourth)
Factorization24 × 32
Divisors1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Greek numeralΡΜΔ´
Roman numeralCXLIV
Binary100100002
Ternary121003
Senary4006
Octal2208
Duodecimal10012

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.[1][2]

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.[3][4] 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 has 16 divisors.[5] 144 is also equal to the sum of the eighth twin prime pair, (71 + 73).[6][7] It is divisible by the value of its φ function, which returns 48 in its case,[8] and there are 21 solutions to the equation ${\displaystyle \varphi (x)=144.}$ This is more than any integer below it, which makes it a highly totient number.[9] In decimal, 144 is the largest of only four sum-product numbers,[10] and it is a Harshad number, where ${\displaystyle 1+4+4=9}$, which divides 144.[11]

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:[12]

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.

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

${\displaystyle 144=12\times 12}$
${\displaystyle 441=21\times 21}$

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

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.[13]

In the Leech lattice

144 is the sum of the divisors of 70: ${\displaystyle \sigma (70)=144}$,[14] 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.[15]: pp.2–11 [16] 144 is relevant in testing whether two vectors in the quaternionic Leech lattice are equivalent under its automorphism group, Conway group ${\displaystyle Co_{0))$: modulo ${\displaystyle 1+i}$, every vector is congruent to either ${\displaystyle 0}$ or a minimal vector that is one of ${\displaystyle 196,560\div {144}=1,365}$ 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.[17][18][19]

References

1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 165
2. ^ Cohn, J. H. E. (1964). "On square Fibonacci numbers". The Journal of the London Mathematical Society. 39: 537–540. doi:10.1112/jlms/s1-39.1.537. MR 0163867.
3. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 165
4. ^ Cohn, J. H. E. (1964). "On square Fibonacci numbers". The Journal of the London Mathematical Society. 39: 537–540. doi:10.1112/jlms/s1-39.1.537. MR 0163867.
5. ^ Sloane, N. J. A. (ed.). "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
6. ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
7. ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
8. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and relatively prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
9. ^ Sloane, N. J. A. (ed.). "Sequence A097942 (Highly totient numbers: each number k on this list has more solutions to the equation phi(x) equal to k than any preceding k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28.
10. ^ Sloane, N. J. A. (ed.). "Sequence A038369 (Numbers k such that k is equal to the product of digits of k by the sum of digits of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
11. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
12. ^ Lander, L. J.; Parkin, T. R. (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72 (6). American Mathematical Society: 1079. doi:10.1090/S0002-9904-1966-11654-3. MR 0197389. S2CID 121274228. Zbl 0145.04903.
13. ^ Sloane, N. J. A. (ed.). "Sequence A003432 (Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
14. ^ Sloane, N. J. A. (ed.). "Sequence A000203 (...the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-06.
15. ^ Slipper, Aaron (2018). Modular magic: The theory of modular forms and the sphere packing problem in dimensions 8 and 24 (PDF) (B.A. thesis). Harvard University. pp. 1–92. S2CID 53005119
16. ^ Sloane, N. J. A. (ed.). "Sequence A351831 (Vector in the 26-dimensional even Lorentzian unimodular lattice II_25,1 used to construct the Leech lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-06.
17. ^ Wilson, Robert A. (1982). "The Quaternionic Lattice for 2G2(4) and its Maximal Subgroups". Journal of Algebra. 77 (2). Elsevier: 451–453. doi:10.1016/0021-8693(82)90266-6. MR 0673128. S2CID 120032380. Zbl 0501.20013.
18. ^ Allcock, Daniel (2005). "Orbits in the Leech Lattice". Experimental Mathematics. 14 (4). Taylor & Francis: 508. doi:10.1080/10586458.2005.10128938. MR 2193810. S2CID 2883584. Zbl 1152.11334.
"The reader should note that each of Wilson’s frames [Wilson 82] contains three of ours, with 3 · 48 = 144 vectors, and has slightly larger stabilizer."
19. ^ Sloane, N. J. A. (ed.). "Sequence A002336 (Maximal kissing number of n-dimensional laminated lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-06.