# Perfect numbers

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Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.

An *aliquot part* of a number is a proper quotient of the number. So for example the aliquot parts of 10 are 1, 2 and 5. These occur since 1 = ^{10}/_{10}, 2 = ^{10}/_{5}, and 5 = ^{10}/_{2}. Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e. a quotient different from the number itself. A perfect number is defined to be one which is equal to the sum of its aliquot parts.

The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.

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

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

8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

*Elements*written around 300BC. It may come as a surprise to many people to learn that there are number theory results in Euclid's

*Elements*since it is thought of as a geometry book. However, although numbers are represented by line segments and so have a geometrical appearance, there are significant number theory results in the

*Elements*. The result which is if interest to us here is Proposition 36 of Book IX of the

*Elements*which states [2]:-

Here 'double proportion' means that each number of the sequence is twice the preceding number. To illustrate this Proposition consider 1 + 2 + 4 = 7 which is prime. ThenIf as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.

which is a perfect number. As a second example, 1 + 2 + 4 + 8 + 16 = 31 which is prime. Then 31 × 16 = 496 which is a perfect number.

Now Euclid gives a rigorous proof of the Proposition and we have the first significant result on perfect numbers. We can restate the Proposition in a slightly more modern form by using the fact, known to the Pythagoreans, that

^{k-1}= 2

^{k}- 1.

*k*> 1, 2

^{k}- 1 is prime then 2

^{k-1}(2

^{k}- 1) is a perfect number.

*Introductio Arithmetica*which gives a classification of numbers based on the concept of perfect numbers. Nicomachus divides numbers into three classes: the superabundant numbers which have the property that the sum of their aliquot parts is greater than the number; deficient numbers which have the property that the sum of their aliquot parts is less than the number; and perfect numbers which have the property that the sum of their aliquot parts is equal to the number (see [8], or [1] for a different translation):-

However Nicomachus has more than number theory in mind for he goes on to show that he is thinking in moral terms in a way that might seem extraordinary to mathematicians today (see [8], or [1] for a different translation):-Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.

Now satisfied with the moral considerations of numbers, Nicomachus goes on to provide biological analogies in which he describes superabundant numbers as being like an animal with (see [8], or [1]):-In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.

Deficient numbers are compared to animals with:-... ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands....

Nicomachus goes on to describe certain results concerning perfect numbers. All of these are given without any attempt at a proof. Let us state them in modern notation.a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue...

*n*th perfect number has

*n*digits.

(2) All perfect numbers are even.

(3) All perfect numbers end in 6 and 8 alternately.

(4) Euclid's algorithm to generate perfect numbers will give all perfect numbers i.e. every perfect number is of the form 2

^{k-1}(2

^{k}- 1), for some

*k*> 1, where 2

^{k}- 1 is prime.

(5) There are infinitely many perfect numbers.

As we have seen this algorithm is precisely that given by Euclid in theThere exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish:1,2,4,8,16,32,64,128,256,512,1024,2048,4096; and then they must be totalled each time there is a new term, and at each totalling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity.

*Elements*. However, it is probable that this methods of generating perfect numbers was part of the general mathematical tradition handed down from before Euclid's time and continuing till Nicomachus wrote his treatise. Whether the five assertions of Nicomachus were based on any more than this algorithm and the fact the there were four perfect numbers known to him 6, 28, 496 and 8128, it is impossible to say, but it does seem unlikely that anything more lies behind the unproved assertions. Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm [1]:-

Despite the fact that Nicomachus offered no justification of his assertions, they were taken as fact for many years. Of course there was the religious significance that we have not mentioned yet, namely that 6 is the number of days taken by God to create the world, and it was believed that the number was chosen by him because it was perfect. Again God chose the next perfect number 28 for the number of days it takes the Moon to travel round the Earth. Saint Augustine (354-430) writes in his famous text... only one is found among the units,6, only one other among the tens,28, and a third in the rank of the hundreds,496alone, and a fourth within the limits of the thousands, that is, below ten thousand,8128. And it is their accompanying characteristic to end alternately in6or8, and always to be even.

When these have been discovered,6among the units and28in the tens, you must do the same to fashion the next. ... the result is496, in the hundreds; and then comes8128in the thousands, and so on, as far as it is convenient for one to follow.

*The City of God*:-

The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote theSix is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect...

*Treatise on amicable numbers*in which he examined when numbers of the form 2

^{n}

*p*, where

*p*is prime, can be perfect. Ibn al-Haytham proved a partial converse to Euclid's proposition in the unpublished work

*Treatise on analysis and synthesis*when he showed that perfect numbers satisfying certain conditions had to be of the form 2

^{k-1}(2

^{k}- 1) where 2

^{k}- 1 is prime.

Among the many Arab mathematicians to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the *Introduction to arithmetic* by Nicomachus. He accepted Nicomachus's classification of numbers but the work is purely mathematical, not containing the moral comments of Nicomachus. Ibn Fallus gave, in his treatise, a table of ten numbers which were claimed to be perfect, the first seven are correct and are in fact the first seven perfect numbers, the remaining three numbers are incorrect. For more details of this impressive work see [6] and [7].

At the beginning of the renaissance of mathematics in Europe around 1500 the assertions of Nicomachus were taken as truths, nothing further being known concerning perfect numbers not even the work of the Arabs. Some even believed the further unjustified and incorrect result that 2^{k-1}(2^{k} - 1) is a perfect number for every odd *k*. Pacioli certainly seems to have believed in this fallacy. Charles de Bovelles, a theologian and philosopher, published a book on perfect numbers in 1509. In it he claimed that Euclid's formula 2^{k-1}(2^{k} - 1) gives a perfect number for all odd integers *k*, see [10]. Yet, rather remarkably, although unknown until comparatively recently, progress had been made.

The fifth perfect number has been discovered again (after the unknown results of the Arabs) and written down in a manuscript dated 1461. It is also in a manuscript which was written by Regiomontanus during his stay at the University of Vienna, which he left in 1461, see [14]. It has also been found in a manuscript written around 1458, while both the fifth and sixth perfect numbers have been found in another manuscript written by the same author probably shortly after 1460. All that is known of this author is that he lived in Florence and was a student of Domenico d'Agostino Vaiaio.

In 1536, Hudalrichus Regius made the first breakthrough which was to become common knowledge to later mathematicians, when he published *Utriusque Arithmetices* in which he gave the factorisation 2^{11} - 1 = 2047 = 23 . 89. With this he had found the first prime p such that 2^{p-1}(2^{p} - 1) is not a perfect number. He also showed that 2^{13} - 1 = 8191 is prime so he had discovered (and made his discovery known) the fifth perfect number 2^{12}(2^{13} - 1) = 33550336. This showed that Nicomachus's first assertion is false since the fifth perfect number has 8 digits. Nicomachus's claim that perfect numbers ended in 6 and 8 alternately still stood however. It is perhaps surprising that Regius, who must have thought he had made one of the major breakthroughs in mathematics, is virtually unheard of today.

J Scheybl gave the sixth perfect number in 1555 in his commentary to a translation of Euclid's *Elements*. This was not noticed until 1977 and therefore did not influence progress on perfect numbers.

The next step forward came in 1603 when Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes). Cataldi was able use his list of primes to show that 2^{17}- 1 = 131071 is prime (since 750^{2} = 562500 > 131071 he could check with a tedious calculation that 131071 had no prime divisors). From this Cataldi now knew the sixth perfect number, namely 2^{16}(2^{17} - 1) = 8589869056. This result by Cataldi showed that Nicomachus's assertion that perfect numbers ended in 6 and 8 alternately was false since the fifth and sixth perfect numbers both ended in 6. Cataldi also used his list of primes to check that 2^{19} - 1 = 524287 was prime (again since 750^{2} = 562500 > 524287) and so he had also found the seventh perfect number, namely 2^{18}(2^{19} - 1) = 137438691328.

As the reader will have already realised, the history of perfect numbers is littered with errors and Cataldi, despite having made the major advance of finding two new perfect numbers, also made some false claims. He writes in *Utriusque Arithmetices* that the exponents *p* = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37 give perfect numbers 2^{p-1}(2^{p} - 1). He is, of course, right for *p* = 2, 3, 5, 7, 13, 17, 19 for which he had a proof from his table of primes, but only one of his further four claims 23, 29, 31, 37 is correct.

Many mathematicians were interested in perfect numbers and tried to contribute to the theory. For example Descartes, in a letter to Mersenne in 1638, wrote [8]:-

The next major contribution was made by Fermat. He told Roberval in 1636 that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic. The treatise would never be written, partly because Fermat never got round to writing his results up properly, but also because he did not achieve the substantial results on perfect numbers he had hoped. In June 1640 Fermat wrote to Mersenne telling him about his discoveries concerning perfect numbers. He wrote:-... I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if22021were prime, in multiplying it by9018009which is a square whose root is composed of the prime numbers3,7,11,13, one would have198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers...

Shortly after writing this letter to Mersenne, Fermat wrote to Frenicle de Bessy on 18 October 1640. In this letter he gave a generalisation of results in the earlier letter stating the result now known as Fermat's Little Theorem which shows that for any prime... here are three propositions I have discovered, upon which I hope to erect a great structure. The numbers less by one than the double progression, like

123456789101112131

371531631272555111023204740958191let them be called the radicals of perfect numbers, since whenever they are prime, they produce them. Put above these numbers in natural progression1,2,3,4,5, etc., which are called their exponents. This done, I say

When the exponent of a radical number is composite, its radical is also composite. Just as6, the exponent of63, is composite, I say that63will be composite.

When the exponent is a prime number, I say that its radical less one is divisible by twice the exponent. Just as7, the exponent of127, is prime, I say that126is a multiple of14.

When the exponent is a prime number, I say that its radical cannot be divisible by any other prime except those that are greater by one than a multiple of double the exponent...

Here are three beautiful propositions which I have found and proved without difficulty, I shall call them the foundations of the invention of perfect numbers. I don't doubt that Frenicle de Bessy got there earlier, but I have only begun and without doubt these propositions will pass as very lovely in the minds of those who have not become sufficiently hypocritical of these matters, and I would be very happy to have the opinion of M Roberval.

*p*and an integer

*a*not divisible by

*p*,

*a*

^{p-1}- 1 is divisible by

*p*. Certainly Fermat found his Little Theorem as a consequence of his investigations into perfect numbers.

Using special cases of his Little Theorem, Fermat was able to disprove two of Cataldi's claims in his June 1640 letter to Mersenne. He showed that 2^{23} - 1 was composite (in fact 2^{23} - 1 = 47 × 178481) and that 2^{37} - 1 was composite (in fact 2^{37} - 1 = 223 × 616318177). Frenicle de Bessy had, earlier in that year, asked Fermat (in correspondence through Mersenne) if there was a perfect number between 10^{20} and 10^{22}. In fact assuming that perfect numbers are of the form 2^{p-1}(2^{p} - 1) where *p* is prime, the question readily translates into asking whether 2^{37} - 1 is prime. Fermat not only states that 2^{37} - 1 is composite in his June 1640 letter, but he tells Mersenne how he factorised it.

Fermat used three theorems:-

*n*is composite, then 2

^{n}- 1 is composite.

(ii) If

*n*is prime, then 2

^{n}- 2 is a multiple of 2

*n*.

(iii) If

*n*is prime,

*p*a prime divisor of 2

^{n}- 1, then

*p*- 1 is a multiple of

*n*.

*p*is a prime divisor of 2

^{37}- 1, then 37 divides

*p*- 1. As

*p*is odd, it is a prime of the form 2 × 37

*m*+1, for some

*m*. The first case to try is

*p*= 149 and this fails (a test division is carried out). The next case to try is 223 (the case

*m*= 3) which succeeds and 2

^{37}- 1 = 223 × 616318177.

Mersenne was very interested in the results that Fermat sent him on perfect numbers and soon produced a claim of his own which was to fascinate mathematicians for a great many years. In 1644 he published *Cogitata physica mathematica * in which he claimed that 2^{p} - 1 is prime (and so 2^{p-1}(2^{p} - 1) is a perfect number) for

*p*= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257

*p*up to 257. Now certainly Mersenne could not have checked these results and he admitted this himself saying:-

The remarkable fact is that Mersenne did very well if this was no more than a guess. There are 47 primes... to tell if a given number of15or20digits is prime, or not, all time would not suffice for the test.

*p*greater than 19 yet less than 258 for which 2

^{p}- 1 might have been either prime or composite. Mersenne got 42 right and made 5 mistakes. A suggestion as to the rule he used in giving his list is made in [9].

Primes of the form 2^{p}- 1 are called Mersenne primes.

The next person to make a major contribution to the question of perfect numbers was Euler. In 1732 he proved that the eighth perfect number was 2^{30}(2^{31} - 1) = 2305843008139952128. It was the first new perfect number discovered for 125 years. Then in 1738 Euler settled the last of Cataldi's claims when he proved that 2^{29} - 1 was not prime (so Cataldi's guesses had not been very good). It should be noticed (as it was at the time) that Mersenne had been right on both counts, since *p* = 31 appears in his list but *p* = 29 does not.

In two manuscripts which were unpublished during his life, Euler proved the converse of Euclid's result by showing that every even perfect number had to be of the form 2^{p-1}(2^{p} - 1). This verifies the fourth assertion of Nicomachus at least in the case of even numbers. It also leads to an easy proof that all even perfect numbers end in either a 6 or 8 (but not alternately). Euler also tried to make some headway on the problem of whether odd perfect numbers existed. He was able to prove the assertion made by Descartes in his letter to Mersenne in 1638 from which we quoted above. He went a little further and proved that any odd perfect number had to have the form

*n*+1)

^{4k+1}

*b*

^{2}

*n*+1 is prime. However, as with most others whose contribution we have examined, Euler made predictions about perfect numbers which turned out to be wrong. He claimed that 2

^{p-1}(2

^{p}- 1) was perfect for

*p*= 41 and

*p*= 47 but Euler does have the distinction of finding his own error, which he corrected in 1753.

The search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims in *Cogitata physica mathematica.* In fact Euler's results had made many people believe that Mersenne had some undisclosed method which would tell him the correct answer. In fact Euler's perfect number 2^{30}(2^{31} - 1) remained the largest known for over 150 years. Mathematicians such as Peter Barlow wrote in his book *Theory of Numbers * published in 1811, that the perfect number 2^{30}(2^{31} - 1):-

This, of course, turned out to be yet one more false assertion about perfect numbers!... is the greatest that ever will be discovered; for as they are merely curious, without being useful, it is not likely that any person will ever attempt to find one beyond it.

The first error in Mersenne's list was discovered in 1876 by Lucas. He was able to show that 2^{67} - 1 is not a prime although his methods did not allow him to find any factors of it. Lucas was also able to verify that one of the numbers in Mersenne's list was correct when he showed that 2^{127} - 1 is a Mersenne prime and so 2^{126}(2^{127}- 1) is indeed a perfect number. Lucas made another important advance which, as modified by Lehmer in 1930, is the basis of computer searches used today to find Mersenne primes, and so to find perfect numbers. Following the announcement by Lucas that *p* = 127 gave the Mersenne prime 2^{p} - 1, Catalan conjectured that, if *m* = 2^{p} - 1 is prime then 2^{m} - 1 is also prime. This Catalan sequence is 2^{p} - 1 where

*p*= 3, 7, 127, 170141183460469231731687303715884105727, ...

^{p}- 1 for

*p*= 170141183460469231731687303715884105727, is prime is well beyond what is possible.

In 1883 Pervusin showed that 2^{60}(2^{61}- 1) is a perfect number. This was shown independently three years later by Seelhoff. Many mathematicians leapt to defend Mersenne saying that the number 67 in his list was a misprint for 61.

In 1903 Cole managed to factorise 2^{67} - 1, the number shown to be composite by Lucas, but for which no factors were known up to that time. In October 1903 Cole presented a paper *On the factorisation of large numbers* to a meeting of the American Mathematical Society. In one of the strangest 'talks' ever given, Cole wrote on the blackboard

^{67}- 1 = 147573952589676412927.

^{67}- 1 in about a second - times have changed!]

Further mistakes made by Mersenne were found. In 1911 Powers showed that 2^{88}(2^{89} - 1) was a perfect number, then a few years later he showed that 2^{107}- 1 is a prime and so 2^{106}(2^{107}- 1) is a perfect number. In 1922 Kraitchik showed that Mersenne was wrong in his claims for his largest prime of 257 when he showed that 2^{257}- 1 is not prime.

We have followed the progress of finding even perfect numbers but there was also attempts to show that an odd perfect number could not exist. The main thrust of progress here has been to show the minimum number of distinct prime factors that an odd perfect number must have. Sylvester worked on this problem and wrote (see [20]):-

In fact Sylvester proved in 1888 that any odd perfect number must have at least 4 distinct prime factors. Later in the same year he improved his result to five factors and, over the years, this has been steadily improved until today we know that an odd perfect number would have to have at least eight distinct prime factors, and at least 29 prime factors which are not necessarily distinct. It is also known that such a number would have more than 300 digits and a prime divisor greater than 10... the existence of[an 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.

^{6}. The problem of whether an odd perfect number exists, however, remains unsolved.

Today (2018) 50 perfect numbers are known, 2^{88}(2^{89}- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer. In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers. At the moment the largest known Mersenne prime is 2^{77 232 917} - 1 (which is also the largest known prime) and the corresponding largest known perfect number is 2^{77 232 916} (2^{77 232 917} - 1). It was discovered in December 2017 and this, the 50th such prime to be discovered, contains more than 23 million digits. If you wonder why we have not included the number in decimal form, then let me say that it contains about 350 times as many characters as this whole article on perfect numbers. Also worth noting is the fact that although this is the 50th to be discovered, it is might not be the 50th largest perfect number as not all smaller cases have been ruled out. See the Official announcement.

**Article by:** *J J O'Connor* and *E F Robertson*

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