54 (number)

Number theory

54 is an abundant number^[1] because the sum of its proper divisors (66),^[2] is greater than itself. Like all multiples of 6,^[3] 54 is equal to some of its proper divisors summed together,^[a] so it is also a semiperfect number.^[4] These proper divisors can be summed in various ways to express all positive integers smaller than 54, so 54 is a practical number as well.^[5] Additionally, as an integer for which the arithmetic mean of all its positive divisors (including itself) is also an integer, 54 is an arithmetic number.^[6]

Trigonometry and the golden ratio

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio.^[7][8] This is because the corresponding interior angle is equal to π/5 radians (or 36 degrees).^[b] If that triangle is isoceles, the relationship with the golden ratio makes it a golden triangle. The golden triangle is most readily found as the spikes on a regular pentagram.

Regular number used in Assyro-Babylonian mathematics

As a regular number, 54 is a divisor of many powers of 60.^[c] This is an important property in Assyro-Babylonian mathematics because that system uses a sexagesimal (base-60) number system. In base 60, the reciprocal of a regular number has a finite representation. Babylonian computers kept tables of these reciprocals to make their work more efficient. Using regular numbers simplifies multiplication and division in base 60 because dividing a by b can be done by multiplying a by b's reciprocal when b is a regular number.^[9][10]

For instance, division by 54 can be achieved in the Assyro-Babylonian system by multiplying by 4000 because 60³ ÷ 54 = 60³ × (1/54) = 4000. In base 60, 4000 can be written as 1:6:40.^[d] Because the Assyro-Babylonian system does not have a symbol separating the fractional and integer parts of a number^[11] and does not have the concept of 0 as a number,^[12] it does not specify the power of the starting digit. Accordingly, 1/54 can also be written as 1:6:40.^[e][11] Therefore, the result of multiplication by 1:6:40 (4000) has the same Assyro-Babylonian representation as the result of multiplication by 1:6:40 (1/54). To convert from the former to the latter, the result's representation is interpreted as a number shifted three base-60 places to the right, reducing it by a factor of 60³.^[f]

Graph theory

The second Ellingham–Horton graph was published by Mark N. Ellingham and Joseph D. Horton in 1983; it is of order 54.^[13] These graphs provided further counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.^[14] Horton disproved the conjecture some years earlier with the Horton graph, but that was larger at 92 vertices.^[15] The smallest known counter-example is now 50 vertices.^[16]