Solitary Numbers
A number that belongs to a singleton club, because no other number is friendly with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary. For a prime number p we have σ(p) = p + 1, which is coprime with p.
No general method is known for determining whether a number is friendly or solitary. The smallest number whose classification is unknown (as of 2009) is 10; it is conjectured to be solitary; if not, its smallest friend is a fairly large number.
Read more about this topic: Friendly Number
Famous quotes containing the words solitary and/or numbers:
“To muse, and spill her solitary Tea,”
—Alexander Pope (1688–1744)
“Our religion vulgarly stands on numbers of believers. Whenever the appeal is made—no matter how indirectly—to numbers, proclamation is then and there made, that religion is not. He that finds God a sweet, enveloping presence, who shall dare to come in?”
—Ralph Waldo Emerson (1803–1882)