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 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 , 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 nnth 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:
Another number that shares this property is
169, where , while
144 is the
sum of the divisors of
70: ,[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 latticeII25,1.[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:
modulo, every vector is congruent to either or a minimal vector that is one of 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]
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.
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.
MR0163867.
^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.
MR0163867.
"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."