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:
“He who is ready to despair in solitary peril, plucks up a heart in the presence of another. In a plurality of comrades is much countenance and consolation.”
—Herman Melville (1819–1891)
“The forward Youth that would appear
Must now forsake his Muses dear,
Nor in the Shadows sing
His Numbers languishing.”
—Andrew Marvell (1621–1678)