Perfect Numbers

For the 2012 film, see Perfect Number (film).

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος αριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby $p\left(p+1\right)/2$ is an even perfect number whenever $p$ is what is now called a Mersenne prime. Much later, Euler proved that all even perfect numbers are of this form.

It is not known whether there are any odd perfect numbers.

Examples

The first perfect number is OEIS).

Discovery

These first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus had noted 8128 as early as 100 AD.[1] In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest reference to a fifth perfect number, with 33,550,336 being correctly identified for the first time.[2][3] In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056)[4] and the seventh (137,438,691,328) perfect numbers.[5]

Even perfect numbers

Euclid proved that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime (Euclid, Prop. IX.36).

For example, the first four perfect numbers are generated by the formula 2p−1(2p−1), with p a prime number, as follows:

for p = 2:   21(22−1) = 6
for p = 3:   22(23−1) = 28
for p = 5:   24(25−1) = 496
for p = 7:   26(27−1) = 8128.

Prime numbers of the form 2p−1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p−1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p−1 with a prime p are prime; for example, 211−1 = 2047 = 23 × 89 is not a prime number. (All factors of 2p−1 will be congruent to 1 mod 2p. For example, 211−1 = 2047 = 23 × 89 and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever p is a Sophie Germain prime—that is, 2p+1 is also prime—and 2p+1 is congruent to 1 or 7 mod 8, then 2p + 1 will be a factor of 2p−1.) In fact, Mersenne primes are very rare—of the 1,622,441 prime numbers p up to 25,964,951, [6] [7] 2p−1 is prime for only 42 of them.

Over a millennium after Euclid, Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of the form 2p−1(2p−1) where 2p−1 is prime, but he was not able to prove this result.[8] It was not until the 18th century that Leonhard Euler proved that the formula 2p−1(2p−1) will yield all the even perfect numbers. Thus, there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler Theorem. As of February 2013, 48 Mersenne primes are known,[9] and therefore 48 even perfect numbers (the largest of which is 257885160 × (257885161−1) with 34,850,340 digits).

The first 42 even perfect numbers are 2p−1(2p−1) for

p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, and 25964951 (sequence

The other 6 known are for p = 30402457, 32582657, 37156667, 42643801, 43112609, and 57885161. It is not known whether there are (or are not) any other perfect numbers besides the ones known after the 42nd. Neither is it known whether there are infinitely many Mersenne primes; this is one of the unsolved problems in mathematics. The search for new Mersenne primes is the goal of the GIMPS distributed computing project.

As well as having the form 2p−1(2p−1), each even perfect number is the (2p−1)th triangular number (and hence equal to the sum of the integers from 1 to 2p−1) and the 2p−1th hexagonal number. Furthermore, each even perfect number except for 6 is the ((2p+1)/3)th centered nonagonal number and is equal to the sum of the first 2(p−1)/2 odd cubes:



\begin{align} 6 & = 2^1(2^2-1) & & = 1+2+3, \\[8pt] 28 & = 2^2(2^3-1) & & = 1+2+3+4+5+6+7 = 1^3+3^3, \\[8pt] 496 & = 2^4(2^5-1) & & = 1+2+3+\cdots+29+30+31 \\ & & & = 1^3+3^3+5^3+7^3, \\[8pt] 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, \\[8pt] 33550336 & = 2^{12}(2^{13}-1) & & = 1+2+3+\cdots+8189+8190+8191 \\ & & & = 1^3+3^3+5^3+\cdots+123^3+125^3+127^3. \end{align}

Even perfect numbers (except 6) are of the form

$T_\left\{2^p - 1\right\} = 1 + \frac\left\{\left(2^p-2\right) \times \left(2^p+1\right)\right\}\left\{2\right\} = 1 + 9 \times T_\left\{\left(2^p - 2\right)/3\right\}$

with each resulting triangular number (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with 3, 55, 903, 3727815, ....[11] This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2p−1(2p−1) with odd prime p and, in fact, with all numbers of the form 2m−1(2m−1) for odd integer (not necessarily prime) m.

Owing to their form, 2p−1(2p−1), every even perfect number is represented in binary as p ones followed by p − 1  zeros:

610 = 1102
2810 = 111002
49610 = 1111100002
812810 = 11111110000002
3355033610 = 11111111111110000000000002.

Thus every even perfect number is a pernicious number.

Note that every even perfect number is also a practical number (c.f. Related concepts).

Odd perfect numbers

 Are there any odd perfect numbers?

It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.

Any odd perfect number N must satisfy the following conditions:

• N > 101500.[12]
• N is not divisible by 105.[13]
• N is of the form N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324).[14]
• N is of the form
$N=q^\left\{\alpha\right\} p_1^\left\{2e_1\right\} \cdots p_k^\left\{2e_k\right\},$
where:
• qp1, ..., pk are distinct primes (Euler).
• q ≡ α ≡ 1 (mod 4) (Euler).
• The smallest prime factor of N is less than (2k + 8) / 3.[15]
• Either qα > 1062, or p j2ej  > 1062 for some j.[12]
• N < 24k+1.[16]
• The largest prime factor of N is greater than 108.[17]
• The second largest prime factor is greater than 104, and the third largest prime factor is greater than 100.[18][19]
• N has at least 101 prime factors and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.[12][20]

In 1888, Sylvester stated:[21]

...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.

Minor results

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:

• The only even perfect number of the form x3 + 1 is 28 (Makowski 1962).
• 28 is also the only even perfect number that is a sum of two positive integral cubes (Gallardo 2010).[22]
• The reciprocals of the divisors of a perfect number N must 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. (This is particularly easy to see, just by taking the definition of a perfect number, $\sigma_1\left(n\right) = 2n$, and dividing both sides by n.)
• The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.
• The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and a class of numbers formed from Fermat primes in a similar way to the construction of even perfect numbers from Mersenne primes. [23]
• The number of perfect numbers less than n is less than $c\sqrt\left\{n\right\}$, where c > 0 is a constant.[24] In fact it is $o\left(\sqrt\left\{n\right\}\right)$, using little-o notation.[25]
• Every even perfect number ends in 6 or 28, base ten.[11]

Related concepts

The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than 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.

All perfect numbers are also $\mathcal\left\{S\right\}$-perfect numbers, or Granville numbers.

References

• Euclid, D.E. Joyce's website for a translation and discussion of this proposition and its proof.
• H.-J. Kanold, "Untersuchungen über ungerade vollkommene Zahlen", Journal für die Reine und Angewandte Mathematik, 183 (1941), pp. 98–109.
• R. Steuerwald, "Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl", S.-B. Bayer. Akad. Wiss., 1937, pp. 69–72.

• Nankar, M.L.: "History of perfect numbers," Ganita Bharati 1, no. 1–2 (1979), 7–8.
• Hagis, P.: "A Lower Bound for the set of odd Perfect Prime Numbers", Mathematics of Computation 27, (1973), 951–953.
• Riele, H.J.J. "Perfect Numbers and Aliquot Sequences" in H.W. Lenstra and R. Tijdeman (eds.): Computational Methods in Number Theory, Vol. 154, Amsterdam, 1982, pp. 141–157.
• Riesel, H. Prime Numbers and Computer Methods for Factorisation, Birkhauser, 1985.
• Template:Springer
• David Moews: Perfect, amicable and sociable numbers
• Perfect numbers – History and Theory
• MathWorld.
• "On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• OddPerfect.org A projected distributed computing project to search for odd perfect numbers.
• Great Internet Mersenne Prime Search
• Perfect Numbers, math forum at Drexel.

Template:Classes of natural numbers

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