Perfect numbers are Integers such that

(1) |

(2) |

etc.

Perfect numbers are intimately connected with a class of numbers known as Mersenne Primes.
This can be demonstrated by considering a perfect number of the form where is Prime. Then

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

# | ||

1 | 2 | 6 |

2 | 3 | 28 |

3 | 5 | 496 |

4 | 7 | 8128 |

5 | 13 | 33550336 |

6 | 17 | 8589869056 |

7 | 19 | 137438691328 |

8 | 31 | 2305843008139952128 |

All Even perfect numbers are of this form, as was proved by Euler in a posthumous paper. The only even perfect number of the form is 28 (Makowski 1962).

It is not known if any Odd perfect numbers exist, although numbers up to have been checked (Brent *et al. * 1991,
Guy 1994) without success, improving the result of Tuckerman (1973), who checked odd numbers up to . Euler
showed that an Odd perfect number, if it exists, must be of the form

(10) |

Every perfect number of the form
can be written

(11) |

(12) | |||

(13) | |||

(14) |

(Singh 1997). All Even perfect numbers are of the form

(15) |

(16) |

(17) |

(18) |

(19) |

If , is said to be an Abundant Number. If , is said to be a Deficient Number. And if for a Positive Integer , is said to be a Multiperfect Number of order .

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, pp. 66-67, 1987.

Brent, R. P.; Cohen, G. L. L.; and te Riele, H. J. J. ``Improved Techniques for Lower Bounds for Odd Perfect Numbers.''
*Math. Comput.* **57**, 857-868, 1991.

Conway, J. H. and Guy, R. K. ``Perfect Numbers.'' In *The Book of Numbers.* New York: Springer-Verlag, pp. 136-137, 1996.

Dickson, L. E. ``Notes on the Theory of Numbers.'' *Amer. Math. Monthly* **18**, 109-111, 1911.

Dickson, L. E. *History of the Theory of Numbers, Vol. 1: Divisibility and Primality.* New York: Chelsea,
pp. 3-33, 1952.

Dunham, W. *Journey Through Genius: The Great Theorems of Mathematics.* New York: Wiley, p. 75, 1990.

Eaton, C. F. ``Problem 1482.'' *Math. Mag.* **68**, 307, 1995.

Eaton, C. F. ``Perfect Number in Terms of Triangular Numbers.'' Solution to Problem 1482. *Math. Mag.* **69**, 308-309, 1996.

Gardner, M. ``Perfect, Amicable, Sociable.'' Ch. 12 in
*Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American.* New York: Vintage,
pp. 160-171, 1978.

Guy, R. K. ``Perfect Numbers.'' §B1 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, p. 145, 1994.

Kraitchik, M. ``Mersenne Numbers and Perfect Numbers.'' §3.5 in *Mathematical Recreations.* New York: W. W. Norton,
pp. 70-73, 1942.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, pp. 145 and 147-151, 1979.

Makowski, A. ``Remark on Perfect Numbers.'' *Elemente Math.* **17**, 109, 1962.

Powers, R. E. ``The Tenth Perfect Number.'' *Amer. Math. Monthly* **18**, 195-196, 1911.

Shanks, D. *Solved and Unsolved Problems in Number Theory, 4th ed.* New York: Chelsea, pp. 1-13 and 25-29, 1993.

Singh, S. *Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.*
New York: Walker, pp. 11-13, 1997.

Sloane, N. J. A. Sequence
A000396/M4186
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Tuckerman, B. ``Odd Perfect Numbers: A Search Procedure, and a New Lower Bound of .'' *Not. Amer. Math. Soc.* **15**, 226, 1968.

Tuckerman, B. ``A Search Procedure and Lower Bound for Odd Perfect Numbers.'' *Math. Comp.* **27**, 943-949, 1973.

Zachariou, A. and Zachariou, E. ``Perfect, Semi-Perfect and Ore Numbers.'' *Bull. Soc. Math. Grèce (New Ser.)* **13**, 12-22, 1972.

© 1996-9

1999-05-26