# four hundred ninety-six

The third perfect number, four hundred ninety-six (496) has been known since antiquity. With just one larger perfect number known to him, Euclid was able to prove that all even perfect numbers are the product of a Mersenne prime and the nearest smaller power of two. In the case of 496, these are 31 and 16.

As a counterexample, 496 disproves Thomas Greenwood’s conjecture that an even triangular number with a prime index is one less than a prime, since although 496 is the 31st triangular number, 497 is not a prime.

496 is an important number in physics, and specifically string theory. “The massless bosonic states in this theory consist of a symmetric rank two field, an anti-symmetric rank two field, a scalar field known as the dilaton and a set of 496 gauge fields filling up the adjoint representation of the gauge group $E_{8}\times E_{8}$.” (Sen, 1998) This discovery of the importance of 496, by Michael Green and John Schwartz is credited with ushering in an era of important revelations in string theory.

## References

• 1 D. Wells The Dictionary of Curious and Interesting Numbers Suffolk: Penguin Books (1987): 155
• 2 A. Sen “An Introduction to Non-perturbative String Theory” http://arxiv.org/abs/hep-th/9802051v1ArXiv preprint
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