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:
“I am not solitary whilst I read and write, though nobody is with me. But if a man would be alone, let him look at the stars.”
—Ralph Waldo Emerson (1803–1882)
“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)