Four hundred [and] ninety-six is the natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

 following four hundred [and] ninety-five and preceding four hundred [and] ninety-seven.

In mathematics

496 is most notable for being a perfect number

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime

Mersenne prime

In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

 31, 2⁵ - 1, with 2⁴ ( 2⁵ - 1 ) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number

Harmonic divisor number

In mathematics, a harmonic divisor number, or Ore number , is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are...

, since the number of proper divisors of 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

Harmonic mean

In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

), yields an integer, 5 in this case.

A triangular number

Triangular number

A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

 and a hexagonal number

Hexagonal number

A hexagonal number is a figurate number. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.The formula for the nth hexagonal number...

, 496 is also a centered nonagonal number

Centered nonagonal number

A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers...

 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

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. It is the largest happy number

Happy number

A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 , or it loops endlessly in a cycle which does not include 1...

 less than 500.

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

Nontotient

In number theory, a nontotient is a positive integer n which is not in the range of Euler's totient function φ, that is, for which φ = n has no solution. In other words, n is a nontotient if there is no integer x that has exactly n coprimes below it. All odd numbers are nontotients, except 1,...

.

E₈

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

 has real dimension 496.

In physics

The number 496 is a very important number in superstring theory

Superstring theory

Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...

. In 1984, Michael Green

Michael Green (physicist)

Michael Boris Green FRS is a British physicist and one of the pioneers of string theory. Currently a professor in the Department of Applied Mathematics and Theoretical Physics and a Fellow in Clare Hall at the University of Cambridge in England, he succeeded Stephen Hawking on 1 November 2009...

 and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

 of the gauge group of type I string theory

Type I string theory

In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...

 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

Heterotic string

In physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...

 can admit another possible gauge group, namely E₈ x E₈

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

.