Editing Perfect number (section)

== Even perfect numbers ==
{{See also|Euclid–Euler theorem}}
{{Unsolved|mathematics|Are there infinitely many perfect numbers?}}
[[Euclid]] proved that <math>2^{p-1}(2^p-1)</math> is an even perfect number whenever <math>2^p-1</math> is prime (''[[Euclid's Elements|Elements]]'', Prop. IX.36).

For example, the first four perfect numbers are generated by the formula <math>2^{p-1}(2^p-1),</math> with {{mvar|p}} a [[prime number]], as follows:
<math display=block>\begin{align}
p = 2 &: \quad 2^1(2^2 - 1) = 2 \times 3 = 6 \\
p = 3 &: \quad 2^2(2^3 - 1) = 4 \times 7 = 28 \\
p = 5 &: \quad 2^4(2^5 - 1) = 16 \times 31 = 496 \\
p = 7 &: \quad  2^6(2^7 - 1) = 64 \times 127 = 8128.
\end{align}</math>

Prime numbers of the form <math>2^p-1</math> are known as [[Mersenne prime]]s, after the seventeenth-century monk [[Marin Mersenne]], who studied [[number theory]] and perfect numbers. For <math>2^p-1</math> to be prime, it is necessary that {{mvar|p}} itself be prime. However, not all numbers of the form <math>2^p-1</math> with a prime {{mvar|p}} are prime; for example, {{nowrap|1=2{{sup|11}} − 1 = 2047 = 23 × 89}} is not a prime number.{{efn|All factors of <math>2^p-1</math> are congruent to {{math|1 [[Modular arithmetic|mod]] 2''p''}}. For example, {{nowrap|1=2{{sup|11}} − 1 = 2047 = 23 × 89}}, and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever {{mvar|p}} is a [[Sophie Germain prime]]—that is, {{math|2''p'' + 1}} is also prime—and {{math|2''p'' + 1}} is congruent to 1 or 7 mod 8, then {{math|2''p'' + 1}} will be a factor of <math>2^p-1,</math> which is the case for {{nowrap|1={{mvar|p}} = 11, 23, 83, 131, 179, 191, 239, 251, ...}} {{oeis|id=A002515}}.}} In fact, Mersenne primes are very rare: of the approximately 4 million primes {{mvar|p}} up to 68,874,199, <math>2^p-1</math> is prime for only 48 of them.<ref name="GIMPS Milestones">{{Cite web |title=GIMPS Milestones Report |url=https://www.mersenne.org/report_milestones/ |access-date=28 July 2024 |website=[[Great Internet Mersenne Prime Search]]}}</ref>

While [[Nicomachus]] had stated (without proof) that {{em|all}} perfect numbers were of the form <math>2^{n-1}(2^n-1)</math> where <math>2^n-1</math> is prime (though he stated this somewhat differently), [[Ibn al-Haytham]] (Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref> It was not until the 18th century that [[Leonhard Euler]] proved that the formula <math>2^{p-1}(2^p-1)</math> will yield all the even perfect numbers. Thus, there is a [[bijection|one-to-one correspondence]] 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]].

An exhaustive search by the [[GIMPS]] distributed computing project has shown that the first 48 even perfect numbers are <math>2^{p-1}(2^p-1)</math> for
: {{mvar|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, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609 and 57885161 {{OEIS|id=A000043}}.<ref name="GIMPS Milestones" />

Four higher perfect numbers have also been discovered, namely those for which {{mvar|p}} = 74207281, 77232917, 82589933 and 136279841. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for {{mvar|p}} below 109332539. {{As of|2024|10}}, 52 Mersenne primes are known,<ref name="mersenne">{{cite web |url=http://www.mersenne.org/ |title=GIMPS Home |publisher=Mersenne.org |access-date=2024-10-21}}</ref> and therefore 52 even perfect numbers (the largest of which is {{nowrap|2<sup>136279840</sup> × (2<sup>136279841</sup> − 1)}} with 82,048,640 digits). It is [[List of unsolved problems in mathematics|not known]] whether there are [[infinite set|infinitely many]] perfect numbers, nor whether there are infinitely many Mersenne primes.

As well as having the form <math>2^{p-1}(2^p-1)</math>, each even perfect number is the <math>(2^p-1)</math>-th [[triangular number]] (and hence equal to the sum of the integers from 1 to <math>2^p-1</math>) and the <math>2^{p-1}</math>-th [[hexagonal number]]. Furthermore, each even perfect number except for 6 is the <math>\tfrac{2^p+1}{3}</math>-th [[centered nonagonal number]] and is equal to the sum of the first <math>2^\frac{p-1}{2}</math> odd cubes (odd cubes up to the cube of <math>2^\frac{p+1}{2}-1</math>):

<math display=block>\begin{alignat}{3}
  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{alignat}</math>

Even perfect numbers (except 6) are of the form
<math display=block>T_{2^p - 1} = 1 + \frac{(2^p - 2) \times (2^p + 1)}{2} = 1 + 9 \times T_{(2^p - 2)/3}</math>

with each resulting triangular number {{nowrap|T<sub>7</sub> {{=}} 28}}, {{nowrap|T<sub>31</sub> {{=}} 496}}, {{nowrap|T<sub>127</sub> {{=}} 8128}} (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with {{nowrap|T<sub>2</sub> {{=}} 3}}, {{nowrap|T{{sub|10}} {{=}} 55}}, {{nowrap|1=T<sub>42</sub> = 903}}, {{nowrap|1=T<sub>2730</sub> = 3727815, ...}}<ref name="mathworld">{{Mathworld|urlname=PerfectNumber|title=Perfect Number}}</ref> It follows that by 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 {{nowrap|1=8 + 1 + 2 + 8 = 19}}, {{nowrap|1=1 + 9 = 10}}, and {{nowrap|1=1 + 0 = 1}}. This works with all perfect numbers <math>2^{p-1}(2^p-1)</math> with odd prime {{mvar|p}} and, in fact, with {{em|all}} numbers of the form <math>2^{m-1}(2^m-1)</math> for odd integer (not necessarily prime) {{mvar|m}}.

Owing to their form, <math>2^{p-1}(2^p-1),</math> every even perfect number is represented in binary form as {{mvar|p}} ones followed by {{math|''p'' − 1}} zeros; for example:

<math display=block>\begin{array}{rcl}
6_{10} =& 2^2 + 2^1 &= 110_2 \\
28_{10} =& 2^4 + 2^3 + 2^2 &= 11100_2 \\
496_{10} =& 2^8 + 2^7 + 2^6 + 2^5 + 2^4 &= 111110000_2 \\
8128_{10} =& \!\! 2^{12} + 2^{11} + 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 \!\! &= 1111111000000_2
\end{array}</math>

Thus every even perfect number is a [[pernicious number]].

Every even perfect number is also a [[practical number]] (cf. [[#Related concepts|Related concepts]]).