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 love all waste
And solitary places; where we taste
The pleasure of believing what we see
Is boundless, as we wish our souls to be.”
—Percy Bysshe Shelley (1792–1822)
“And when all bodies meet
In Lethe to be drowned,
Then only numbers sweet
With endless life are crowned.”
—Robert Herrick (1591–1674)