144 (number)
| ||||
|---|---|---|---|---|
| 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 | |||
| Quaternary | 21004 | |||
| Quinary | 10345 | |||
| Senary | 4006 | |||
| Octal | 2208 | |||
| Duodecimal | 10012 | |||
| Hexadecimal | 9016 | |||
| Vigesimal | 7420 | |||
| Base 36 | 4036 | |||
144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145. It is a dozen dozens, or one gross.
In 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.[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]
L. J. Lander and T. R. Parkin, p. 1079
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]
In sports
- College Hoops Net (CHN) annual ranking of the Top 144 NCAA college basketball teams in 144 days.[17]
- The CFL record for career touchdown receptions, held by Milt Stegall of the Winnipeg Blue Bombers.
In other fields
144 is also:
- The year AD 144 or 144 BC.
- 144 Vibilia is a dark, large Main belt asteroid.
- The measurement, in cubits, of the wall of New Jerusalem shown by the seventh angel (Bible, Revelation 21:17).
- Psalm 144
- The atomic number of unquadquadium, a temporary chemical element.
- The Intel 8086 instruction for no operation (NOP).
- Sonnet 144 by William Shakespeare.
- Tupolev Tu-144 was the first supersonic transport aircraft (SST), constructed under the direction of the Soviet Tupolev.
- 1:144 scale is a scale used for some scale models.
- The number of square inches in a square foot.
- Mahjong is usually played with a set of 144 tiles.
- 144 (film), 2015 Indian Tamil film.
See also
- List of highways numbered 144
References
- ↑ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 165
- ↑ 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.
- ↑ Sloane, N. J. A., ed. "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". OEIS Foundation. https://oeis.org/A000005. Retrieved 2023-04-04.
- ↑ Sloane, N. J. A., ed. "Sequence A001359 (Lesser of twin primes.)". OEIS Foundation. https://oeis.org/A001359. Retrieved 2023-04-04.
- ↑ Sloane, N. J. A., ed. "Sequence A006512 (Greater of twin primes.)". OEIS Foundation. https://oeis.org/A006512. Retrieved 2023-04-04.
- ↑ 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.)". OEIS Foundation. https://oeis.org/A000010. Retrieved 2023-04-04.
- ↑ 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.)". OEIS Foundation. https://oeis.org/A097942. Retrieved 2016-05-28.
- ↑ 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.)". OEIS Foundation. https://oeis.org/A038369. Retrieved 2023-04-04.
- ↑ Sloane, N. J. A., ed. "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". OEIS Foundation. https://oeis.org/A005349. Retrieved 2023-04-04.
- ↑ Lander, L. J.; Parkin, T. R. (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. (American Mathematical Society) 72 (6): 1079. doi:10.1090/S0002-9904-1966-11654-3.
- ↑ Sloane, N. J. A., ed. "Sequence A003432 (Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.)". OEIS Foundation. https://oeis.org/A003432. Retrieved 2023-04-04.
- ↑ Sloane, N. J. A., ed. "Sequence A000203 (...the sum of the divisors of n.)". OEIS Foundation. https://oeis.org/A000203. Retrieved 2023-04-06.
- ↑ 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.)". OEIS Foundation. https://oeis.org/A351831. Retrieved 2023-04-06.
- ↑ Wilson, Robert A. (1982). "The Quaternionic Lattice for 2G2(4) and its Maximal Subgroups". Journal of Algebra (Elsevier) 77 (2): 451–453. doi:10.1016/0021-8693(82)90266-6.
- ↑ Allcock, Daniel (2005). "Orbits in the Leech Lattice". Experimental Mathematics (Taylor & Francis) 14 (4): 508. doi:10.1080/10586458.2005.10128938. https://projecteuclid.org/journals/experimental-mathematics/volume-14/issue-4/Orbits-in-the-Leech-Lattice/em/1136926978.full.
- "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."
- ↑ Sloane, N. J. A., ed. "Sequence A002336 (Maximal kissing number of n-dimensional laminated lattice.)". OEIS Foundation. https://oeis.org/A002336. Retrieved 2023-04-06.
- ↑ College Hoops Net
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Group. (1987): 139–140.
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