496 (number)

From Wikipedia, the free encyclopedia

Four hundred [and] ninety-six is the natural number following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven.

496
List of numbers — Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	496
Ordinal	496th
Factorization	$496 = 2^4 \cdot 31$
Roman numeral	CDXCVI
Binary	111110000
Hexadecimal	1F0

[edit] In mathematics

496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 2⁵ - 1, with 2⁴ ( 2⁵ - 1 ) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.

A triangular number and a hexagonal number, 496 is also a centered nonagonal number and a centered 11-gonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even indexed triangular number is a prime number.

There is no solution to the equation φ(x) = 496, making 496 a nontotient.

E₈ has real dimension 496.

[edit] In physics

The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E₈ x E₈.