144 (number)

144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145. It is coincidentally both the square of twelve (a dozen dozens, or one gross.) and the twelfth Fibonacci number, and the only nontrivial number in the sequence that is square.^[1][2]

← 143 144 145 →
← 140 141 142 143 144 145 146 147 148 149 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
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

Mathematics

144 is a highly totient number.^[3]

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:^[4]

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 other fields

• 1:144 scale is a scale used for some scale models.
• Mahjong is usually played with a set of 144 tiles.

  

  A traditional set of 144 Chinese Mahjong 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.
• 144 is the number of square inches in a square foot.

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. 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.
4. 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.

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Group. (1987): 139–140.

External links

• The Natural Number 144