Perfect Number

Perfect numbers are Integers $n$ such that

$$n=s(n),$$ (1)

where $s(n)$ is the Restricted Divisor Function (i.e., the Sum of Proper Divisors of $n$), or equivalently

$$\sigma(n)=2n,$$ (2)

where $\sigma(n)$ is the Divisor Function (i.e., the Sum of Divisors of $n$ including $n$ itself). The first few perfect numbers are 6, 28, 496, 8128, ... (Sloane's A000396). This follows from the fact that

$$6 = 1+2+3$$

$$28 = 1+2+4+7+14$$

$$496 = 1+2+4+8+16+31+62+124+248,$$

etc.

Perfect numbers are intimately connected with a class of numbers known as Mersenne Primes. This can be demonstrated by considering a perfect number $P$ of the form $P=q 2^{p-1}$ where $q$ is Prime. Then

$$\sigma(P)=2P,$$ (3)

and using

$$\sigma(q)=q+1$$ (4)

for $q$ prime, and

$$\sigma(2^\alpha)=2^{\alpha+1}-1$$ (5)

gives

$$\sigma(q2^{p-1}) = \sigma(q)\sigma(2^{p-1})=(q+1)(2^p-1)$$

$$= 2q2^{p-1}=q2^p$$ (6)

$$q(2^p-1)+2^p-1=q2^p$$ (7)

$$q=2^p-1.$$ (8)

Therefore, if $M_p\equiv q=2^p-1$ is Prime, then

$$P = {\textstyle{1\over 2}}(M_p+1)M_p=2^{p-1}(2^p-1)$$ (9)

is a perfect number, as was stated in Proposition IX.36 of Euclid's Elements (Dunham 1990). The first few perfect numbers are summarized in the following table.

#	$p$	$P$
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 $x^3+1$ is 28 (Makowski 1962).

It is not known if any Odd perfect numbers exist, although numbers up to $10^{300}$ have been checked (Brent et al. 1991, Guy 1994) without success, improving the result of Tuckerman (1973), who checked odd numbers up to $10^{36}$. Euler showed that an Odd perfect number, if it exists, must be of the form

$$m=p^{4a+1}Q^2,$$ (10)

where $p$ is an Odd Prime Relatively Prime to $Q$. In 1887, Sylvester conjectured and in 1925, Gradshtein proved that any Odd perfect number must have at least six different prime aliquot factors (or eight if it is not divisible by 3; Ball and Coxeter 1987). Catalan (1888) proved that if an Odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime aliquot factors. Stuyvaert (1896) proved that an Odd perfect number must be a sum of squares. All Even perfect numbers end in 16, 28, 36, 56, or 76 (Lucas 1891) and, with the exception of 6, have Digital Root 1.

Every perfect number of the form $2^p(2^{p+1}-1)$ can be written

$$2^p(2^{p+1}-1)=\sum_{k=1}^{p/2} (2k-1)^3.$$ (11)

All perfect numbers are Hexagonal Numbers and therefore Triangular Numbers. It therefore follows that perfect numbers are always the sum of consecutive Positive integers starting at 1, for example,

$$6 = \sum_{n=1}^3 n$$ (12)

$$28 = \sum_{n=1}^7 n$$ (13)

$$496 = \sum_{n=1}^{31} n$$ (14)

(Singh 1997). All Even perfect numbers $P>6$ are of the form

$$P+1+9T_n,$$ (15)

where $T_n$ is a Triangular Number

$$T_n={\textstyle{1\over 2}}n(n+1)$$ (16)

such that $n=8j+2$ (Eaton 1995, 1996). The sum of reciprocals of all the divisors of a perfect number is 2, since

$$\underbrace{n+\ldots+c+b+a}_n = 2n$$ (17)

$${n\over a}+{n\over b}+\ldots=2n$$ (18)

$${1\over a}+{1\over b}+\ldots=2.$$ (19)

If $s(n)>n$, $n$ is said to be an Abundant Number. If $s(n)<n$, $n$ is said to be a Deficient Number. And if $s(n)=kn$ for a Positive Integer $k>1$, $n$ is said to be a Multiperfect Number of order $k$.

See also Abundant Number, Aliquot Sequence, Amicable Numbers, Deficient Number, Divisor Function, e-Perfect Number, Harmonic Number, Hyperperfect Number, Infinary Perfect Number, Mersenne Number, Mersenne Prime, Multiperfect Number, Multiplicative Perfect Number, Pluperfect Number, Pseudoperfect Number, Quasiperfect Number, Semiperfect Number, Smith Number, Sociable Numbers, Sublime Number, Superperfect Number, Unitary Perfect Number, Weird Number

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 $10^{36}$.'' 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.