Perfect number

In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself.

Six (6) is the first perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in the OEIS).

These first four perfect numbers were the only ones known to the ancient Greeks.

Even perfect numbers

Euclid discovered that the first four perfect numbers are generated by the formula 2ⁿ⁻¹(2ⁿ − 1):

for n = 2:   2¹(2² − 1) = 6

for n = 3:   2²(2³ − 1) = 28

for n = 5:   2⁴(2⁵ − 1) = 496

for n = 7:   2⁶(2⁷ − 1) = 8128

Noticing that 2ⁿ − 1 is a prime number in each instance, Euclid proved that the formula 2ⁿ⁻¹(2ⁿ − 1) gives an even perfect number whenever 2ⁿ − 1 is prime.

Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2¹¹ − 1 = 2047 = 23 · 89 is not prime and therefore n = 11 does not yield a perfect number. Two other wrong assumptions were:

• The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively.
• The perfect numbers would alternately end in 6 or 8.

The fifth perfect number ($33550336=2^{12}(2^{13}-1)$) has 8 digits, thus debunking the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It has been shown that the last digit of any even perfect number must be 6 or 8.

In order for $2^{n}-1$ to be prime, it is necessary that $n$ should be prime. Prime numbers of the form 2ⁿ − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.

Two millennia after Euclid, Euler proved that the formula 2ⁿ⁻¹(2ⁿ − 1) will yield all the even perfect numbers. Thus, every Mersenne prime will yield a distinct even perfect number—there is a concrete one-to-one association between even perfect numbers and Mersenne primes. This result is often referred to as the "Euclid-Euler Theorem."

Only finitely many Mersenne primes are presently known, and it is unknown whether there are infinitely many of them. Thus it also remains uncertain whether there are infinitely many even perfect numbers.

Every even perfect number is a triangular number. Since any even perfect number has the form 2ⁿ⁻¹(2ⁿ − 1), it is the sum of all natural numbers up to 2ⁿ − 1. This follows from the general formula stating that the sum of the first m positive integers equals (m² + m)/2. Furthermore, any even perfect number except the first one is the sum of the first 2^(n−1)/2 odd cubes:

$6=2^{1}(2^{2}-1)=1+2+3,\,$

$28=2^{2}(2^{3}-1)=1+2+3+4+5+6+7=1^{3}+3^{3},\,$

$496=2^{4}(2^{5}-1)=1+2+3+\cdots +29+30+31=1^{3}+3^{3}+5^{3}+7^{3},\,$

$8128=2^{6}(2^{7}-1)=1+2+3+\cdots +125+126+127=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}.\,$

Another interesting fact is that the reciprocals of the factors of a perfect number add up to 2:

• For 6, we have $1/6+1/3+1/2+1/1=2$;
• For 28, we have $1/28+1/14+1/7+1/4+1/2+1/1=2$, etc.

Odd perfect numbers

It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that have helped to locate one or otherwise resolve the question of their existence. Recently, Carl Pomerance and Joshua Zelinsky have both presented heuristics which strongly suggest that no odd perfect numbers exist.

Any odd perfect number N must satisfy the following conditions:

• N is of the form

$N=q^{\alpha }p_{1}^{2e_{1}}\ldots p_{k}^{2e_{k}},$

where q, p₁, …, p_k are distinct primes and q ≡ α ≡ 1 (mod 4) (Euler).

• N is bigger than 10³⁰⁰.
• N is of the form 4j + 1 (A. Stern, 1896)
• N has at least 8 distinct prime factors (and at least 11 if it is not divisible by 3) (Peter Hagis).
• N has at least 75 prime factors in total, including repetitions (Kevin Hare, 2005).
• N has at least one prime factor greater than 10⁷, two prime factors greater than 10⁴, and three prime factors greater than 100.
• N is less than $2^{4^{n}}$ where n=k+1 is the number of distinct prime factors.
• N is of the form 12j + 1 or 36j + 9 (Jacques Touchard). (An elementary proof was discovered by Judy A. Holdener)

See also

• Semiperfect number
• Quasiperfect number
• Almost perfect number
• Multiply perfect number
• Hyperperfect number
• Unitary perfect number

The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than twice the number itself are called deficient, and where it is greater than twice the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.

By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.

References

• Kevin Hare, New techniques for bounds on the total number of prime factors of an odd perfect number. Preprint, 2005. Available from his webpage.

External links

• David Moews: Perfect, amicable and sociable numbers
• Perfect numbers - History and Theory
• Perfect Number - from MathWorld
• List of Perfect Numbers at the On-Line Encyclopedia of Integer Sequences