Large Clubs
It is an open problem whether there are infinitely large clubs of mutually friendly numbers. The perfect numbers form a club, and it is conjectured that there are infinitely many perfect numbers (at least as many as there are Mersenne primes), but no proof is known. As of June 2009, 47 perfect numbers are known, the largest of which has more than 25 million digits in decimal notation. There are clubs with more known members, in particular those formed by multiply perfect numbers, which are numbers whose abundancy is an integer. As of early 2011, the club of friendly numbers with abundancy equal to 9 has 2091 known members. Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.
Read more about this topic: Friendly Number
Famous quotes containing the words large and/or clubs:
“It is from quiet places like this all over the world that the forces accumulate which presently will overbear any attempt to accomplish evil on a large scale. Like the rivulets gathering into the river, and the river into the seas, there come from communities like this streams that fertilize the consciences of men, and it is the conscience of the world that we are trying to place upon the throne which others would usurp.”
—Woodrow Wilson (1856–1924)
“Remember that the peer group is important to young adolescents, and there’s nothing wrong with that. Parents are often just as important, however. Don’t give up on the idea that you can make a difference.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.5 (1985)