Perfect numbers

It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity. It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked. Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties. Before we begin to look at the history of the study of perfect numbers, we define the concepts which are involved.

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.

6 = 1 + 2 + 3,
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

The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid's 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 :

If 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.

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. Then

(the sum) (the last) = 7 4 = 28,

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

1 + 2 + 4 + ... + 2^k-1 = 2^k - 1.

The Proposition now reads:-

If, for some k > 1, 2^k - 1 is prime then 2^k-1(2^k - 1) is a perfect number.

The next significant study of perfect numbers was made by Nicomachus of Gerasa. Around 100 AD Nicomachus wrote his famous text 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):-

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.

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):-

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.

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]):-

... 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....

Deficient numbers are compared to animals with:-

a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue...

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.

(1) The nth 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.

We will see how these assertions have stood the test of time as we carry on with our discussions, but let us say at this point that assertions (1) and (3) are false while, as stated, (2), (4) and (5) are still open questions. However, since the time of Nicomachus we do know a lot more about his five assertions than the simplistic statement we have just made. Let us look in more detail at Nicomachus's description of the algorithm to generate perfect numbers which is assertion (4) above (see [8], or [1]):-

There 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.

As we have seen this algorithm is precisely that given by Euclid in the 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]:-

... only one is found among the units, 6, only one other among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128. And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even.

When these have been discovered, 6 among the units and 28 in the tens, you must do the same to fashion the next. ... the result is 496, in the hundreds; and then comes 8128 in the thousands, and so on, as far as it is convenient for one to follow.

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 The City of God :

Six 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...

The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined when numbers of the form 2ⁿp, 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.

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 2¹¹ - 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 2¹³ - 1 = 8191 is prime so he had discovered (and made his discovery known) the fifth perfect number 2¹²(2¹³ - 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 2¹⁷- 1 = 131071 is prime (since 750² = 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 2¹⁶(2¹⁷ - 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 2¹⁹ - 1 = 524287 was prime (again since 750² = 562500 > 524287) and so he had also found the seventh perfect number, namely 2¹⁸(2¹⁹ - 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 :

... 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, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, 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...

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:-

... 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

1  2  3   4   5   6    7    8    9    10    11    12    13

1  3  7  15  31  63  127  255  511  1023  2047  4095  8191

let them be called the radicals of perfect numbers, since whenever they are prime, they produce them. Put above these numbers in natural progression 1, 2, 3, 4, 5, etc., which are called their exponents. This done, I say

1. When the exponent of a radical number is composite, its radical is also composite. Just as 6, the exponent of 63, is composite, I say that 63 will be composite.
   
2. When the exponent is a prime number, I say that its radical less one is divisible by twice the exponent. Just as 7, the exponent of 127, is prime, I say that 126 is a multiple of 14.
   
3. 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.

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 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 2²³ - 1 was composite (in fact 2²³ - 1 = 47 178481) and that 2³⁷ - 1 was composite (in fact 2³⁷ - 1 = 223 616318177). Frenicle de Bessy had, earlier in that year, asked Fermat (in correspondence through Mersenne) if there was a perfect number between 10²⁰ and 10²². 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 2³⁷ - 1 is prime. Fermat not only states that 2³⁷ - 1 is composite in his June 1640 letter, but he tells Mersenne how he factorised it.

Fermat used three theorems:-

(i) If n is composite, then 2ⁿ - 1 is composite.

(ii) If n is prime, then 2ⁿ - 2 is a multiple of 2n.

(iii) If n is prime, p a prime divisor of 2ⁿ- 1, then p - 1 is a multiple of n.

Note that (i) is trivial while (ii) and (iii) are special cases of Fermat's Little Theorem. Fermat proceeds as follows: If p is a prime divisor of 2³⁷ - 1, then 37 divides p - 1. As p is odd, it is a prime of the form 2 37m+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 2³⁷ - 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

and for no other value of p up to 257. Now certainly Mersenne could not have checked these results and he admitted this himself saying:-

... to tell if a given number of 15 or 20 digits is prime, or not, all time would not suffice for the test.

The remarkable fact is that Mersenne did very well if this was no more than a guess. There are 47 primes 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.

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 2³⁰(2³¹ - 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 2²⁹ - 1 was not prime (so Cataldi's guesses had not been very good). Now 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

(4n+1)^4k+1 b²

where 4n+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 2³⁰(2³¹ - 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 2³⁰(2³¹ - 1):-

... 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.

This, of course, turned out to be yet one more false assertion about perfect numbers!

The first error in Mersenne's list was discovered in 1876 by Lucas. He was able to show that 2⁶⁷ - 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 2¹²⁷ - 1 is a Mersenne prime and so 2¹²⁶(2¹²⁷- 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, ...

Of course if this conjecture were true it would solve the still open question of whether there are an infinite number of Mersenne primes (and also solve the still open question of whether there are infinitely many perfect numbers). However checking whether the fourth term of this sequence, namely 2^p - 1 for p = 170141183460469231731687303715884105727, is prime is well beyond what is possible.

In 1883 Pervusin showed that 2⁶⁰(2⁶¹- 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 2⁶⁷ - 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

2⁶⁷ - 1 = 147573952589676412927.

Then he wrote 761838257287 and underneath it 193707721. Without speaking a work he multiplied the two numbers together to get 147573952589676412927 and sat down to applause from the audience. [It is worth remarking that the computer into which I [EFR] am typing this article gave this factorisation of 2⁶⁷ - 1 in about a second - times have changed!]

Further mistakes made by Mersenne were found. In 1911 Powers showed that 2⁸⁸
(2⁸⁹ - 1) was a perfect number, then a few years later he showed that 2¹⁰¹- 1 is a prime and so 2¹⁰⁰(2¹⁰¹- 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 2²⁵⁷- 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]):-

... 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.

Today 39 perfect numbers are known, 2⁸⁸(2⁸⁹- 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 2^13466917 - 1 (which is also the largest known prime) and the corresponding largest known perfect number is 2^13466916(2^13466917 - 1). It was discovered in December 2001 and this, the 39th such prime to be discovered, contains more than 4 million digits. If you wonder why we haven't included the number in decimal form, then let me say that it contains about 150 times as many characters as this whole article on perfect numbers. Also worth noting is the fact that although this is the 39th to be discovered, it may not be the 39th perfect number as not all smaller cases have been ruled out.

The number Pi

A little known verse of the Bible reads

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)

The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives p = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of ²⁵/₈ = 3.125 and 10 = 3.162 have been traced to much earlier dates: though in defence of Solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. There are some interpretations of this which lead to a much better value.

The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of p including the 'Biblical' value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4(⁸/₉)² = 3.16 as a value for p.

The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation

²²³/₇₁ < p < ²²/₇.

Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that p does not equal ²²/₇, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.p

Here is Archimedes' argument.

Consider a circle of radius 1, in which we inscribe a regular polygon of 3 2^n-1 sides, with semiperimeter b_n, and superscribe a regular polygon of 3 2^n-1 sides, with semiperimeter a_n.

The diagram for the case n = 2 is on the right.

The effect of this procedure is to define an increasing sequence

b₁, b₂, b₃, ...

and a decreasing sequence

a₁, a₂, a₃, ...

such that both sequences have limit p.

Using trigonometrical notation, we see that the two semiperimeters are given by

a_n = K tan(p/K), b_n = K sin(p/K),

where K = 3 2^n-1. Equally, we have

a_n+1 = 2K tan(p/2K), b_n+1 = 2K sin(p/2K),

and it is not a difficult exercise in trigonometry to show that

(1) . . . (1/a_n + 1/b_n) = 2/a_n+1

(2) . . . a_n+1b_n = (b_n+1)².

Archimedes, starting from a₁ = 3 tan(p/3) = 33 and b₁ = 3 sin(p/3) = 33/2, calculated a₂ using (1), then b₂ using (2), then a₃ using (1), then b₃ using (2), and so on until he had calculated a₆ and b₆. His conclusion was that

b₆ < p < a₆.

It is important to realise that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a₆ and b₆ from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.

For of course there is no reason in principle why one should not go on. Various people did, including:

Ptolemy	(c. 150 AD)	3.1416
Zu Chongzhi	(430-501 AD)	355/113
al-Khwarizmi	(c. 800 )	3.1416
al-Kashi	(c. 1430)	14 places
Viète	(1540-1603)	9 places
Roomen	(1561-1615)	17 places
Van Ceulen	(c. 1600)	35 places

Except for Zu Chongzhi, about whom next to nothing is known and who is very unlikely to have known about Archimedes' work, there was no theoretical progress involved in these improvements, only greater stamina in calculation. Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.

Al-Khwarizmi lived in Baghdad, and incidentally gave his name to 'algorithm', while the words al jabr in the title of one of his books gave us the word 'algebra'. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi, one need hardly add, lived in China.

The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for p. One of the earliest was that of Wallis (1616-1703)

2/p = (1.3.3.5.5.7....)/(2.2.4.4.6.6. ...)

and one of the best-known is

p/₄ = 1 - ¹/₃ + ¹/₅ - ¹/₇ + ....

This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).

These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while p arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.

From the point of view of the calculation of p, however, neither is of any use at all. In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = ¹/₂₀₀₀₀, and so we need about 10000 terms of the series. However, Gregory also showed the more general result

(3) . . . tan^-1 x = x - x³/3 + x⁵/5 - ... (-1 x 1)

from which the first series results if we put x = 1. So using the fact that

tan^-1(¹/₃) = p/₆ we get

p/₆ = (¹/₃)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...

which converges much more quickly. The 10^th term is ¹/₁₉ 3⁹3, which is less than 0.00005, and so we have at least 4 places correct after just 9 terms.

An even better idea is to take the formula

(4) . . . p/₄ = tan^-1(¹/₂) + tan^-1(¹/₃)

and then calculate the two series obtained by putting first ¹/₂ and the ¹/₃ into (3).

Clearly we shall get very rapid convergence indeed if we can find a formula something like

p/₄ = tan^-1(¹/_a) + tan^-1(¹/_b)

with a and b large. In 1706 Machin found such a formula:

(5) . . . p/₄ = 4 tan^-1(¹/₅) - tan^-1(¹/₂₃₉)

Actually this is not at all hard to prove, if you know how to prove (4) then there is no real extra difficulty about (5), except that the arithmetic is worse. Thinking it up in the first place is, of course, quite another matter.

With a formula like this available the only difficulty in computing p is the sheer boredom of continuing the calculation. Needless to say, a few people were silly enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit. One of them, an Englishman named Shanks, used Machin's formula to calculate p to 707 places, publishing the results of many years of labour in 1873. Shanks has achieved immortality for a very curious reason which we shall explain in a moment.
Here is a summary of how the improvement went:

1699:	Sharp used Gregory's result to get 71 correct digits
1701:	Machin used an improvement to get 100 digits and the following used his methods:
1719:	de Lagny found 112 correct digits
1789:	Vega got 126 places and in 1794 got 136
1841:	Rutherford calculated 152 digits and in 1853 got 440
1873:	Shanks calculated 707 places of which 527 were correct

Shanks knew that p was irrational since this had been proved in 1761 by Lambert. Shortly after Shanks' calculation it was shown by Lindemann that p is transcendental, that is, p is not the solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that 'squaring the circle' is impossible. The transcendentality of p implies that there is no ruler and compass construction to construct a square equal in area to a given circle.

Very soon after Shanks' calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7's. He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson discovered that Shanks had made an error in the 528^th place, after which all his digits were wrong. In 1949 a computer was used to calculate p to 2000 places. In this and all subsequent computer expansions the number of 7's does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.

You can see 2000 places of p.

We should say a little of how the notation p arose. Oughtred in 1647 used the symbol d/p for the ratio of the diameter of a circle to its circumference. David Gregory (1697) used p/r for the ratio of the circumference of a circle to its radius. The first to use p with its present meaning was an Welsh mathematician William Jones in 1706 when he states 3.14159 andc. = p. Euler adopted the symbol in 1737 and it quickly became a standard notation.

We conclude with one further statistical curiosity about the calculation of p, namely Buffon's needle experiment. If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/p. Various people have tried to calculate p by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got

p = ³⁵⁵/₁₁₃ = 3.1415929

which, incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously good, and the game is given away by the strange number 34080 of tosses. Kendall and Moran comment that a good value can be obtained by stopping the experiment at an optimal moment. If you set in advance how many throws there are to be then this is a very inaccurate way of computing p. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.

Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once. His estimate for p was thus given by

2 0.7857 / p = ¹/₂

from which he got the highly creditable value of p = 3.1428. He was not being serious!

It is almost unbelievable that a definition of p was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined p in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that p/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau's dismissal:-

Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.

G H Hardy replied immediately to Bieberbach in a published note about the consequences of this un-German definition of p

There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbach's reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.

Not only in Germany did p present problems. In the USA the value of p gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.

Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)

The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!

Open questions about the number p

1. Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in p?

2. Brouwer's question: In the decimal expansion of p, is there a place where a thousand consecutive digits are all zero?

3. Is p simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?

4. Is p normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?

5. Is p normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.

6. Another normal question! We know that p is not rational so there is no point from which the digits will repeat. However, if p is normal then the first million digits 314159265358979... will occur from some point. Even if p is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.
   

As a postscript, here is a mnemonic for the decimal expansion of p. Each successive digit is the number of letters in the corresponding word.

How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard...:

3.14159265358979323846264...

The number Phi is an approximation of the Fibonacci sequence, the higher one goes in the Fibonacci Sequence the closer ones gets to the Golden Ratio.

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1.615

...

233 / 144 = 1.618

The division of any two adjacent numbers gives the amazing Golden number : 1.618.

The Golden Ratio can be expressed as 1.618 and 0.618 and is known as Phi and phi, respectively; phi being the reciprocal of Phi... This is a very unique property that only the Golden Ratio possesses:

1 / Phi = phi  (1 / 1.618 = 0.618)

and...

1 / phi = Phi  (1 / 0.618 = 1.618)

Also, Phi Squared = Phi + 1  (1.618 ^2 = 1.618 + 1)

...and Phi multiplied by phi = 1  (1.618 * 0.618 = 1)

Phi is not a fraction: In other words, there is no way to express Phi as using two integers, e.g. (2/3)

Deriving Phi:

Phi = Square root of 5 + 1 / 2...  or  (5+1)/2

Phi to 31 decimal places: 1.6180339887498948482045868343656

The number e

One of the first articles which we included in the "History Topics" section of our web archive was on the history of p. It is a very popular article and has prompted many to ask for a similar article about the number e. There is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one for p. The number e is, compared to p, a relative newcomer on the mathematics scene.

The number e first comes into mathematics in a very minor way. This was in 1618 when, in an appendix to Napier's work on logarithms, a table appeared giving the natural logarithms of various numbers. However, that these were logarithms to base e was not recognised since the base to which logarithms are computed did not arise in the way that logarithms were thought about at this time. Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. We will come back to this point later in this essay. This table in the appendix, although carrying no author's name, was almost certainly written by Oughtred. A few years later, in 1624, again e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work.

The next possible occurrence of e is again dubious. In 1647 Saint-Vincent computed the area under a rectangular hyperbola. Whether he recognised the connection with logarithms is open to debate, and even if he did there was little reason for him to come across the number e explicitly. Certainly by 1661 Huygens understood the relation between the rectangular hyperbola and the logarithm. He examined explicitly the relation between the area under the rectangular hyperbola yx = 1 and the logarithm. Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1. This is the property that makes e the base of natural logarithms, but this was not understood by mathematicians at this time, although they were slowly approaching such an understanding.

Huygens made another advance in 1661. He defined a curve which he calls "logarithmic" but in our terminology we would refer to it as an exponential curve, having the form y = ka^x. Again out of this comes the logarithm to base 10 of e, which Huygens calculated to 17 decimal places. However, it appears as the calculation of a constant in his work and is not recognised as the logarithm of a number (so again it is a close call but e remains unrecognised).

Further work on logarithms followed which still does not see the number e appear as such, but the work does contribute to the development of logarithms. In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x). In this work Mercator uses the term "natural logarithm" for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just round the corner.

Perhaps surprisingly, since this work on logarithms had come so close to recognising the number e, when e is first "discovered" it is not through the notion of logarithm at all but rather through a study of compound interest. In 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of (1 + ¹/n)ⁿ as n tends to infinity. He used the binomial theorem to show that the limit had to lie between 2 and 3 so we could consider this to be the first approximation found to e. Also if we accept this as a definition of e, it is the first time that a number was defined by a limiting process. He certainly did not recognise any connection between his work and that on logarithms.

We mentioned above that logarithms were not thought of in the early years of their development as having any connection with exponents. Of course from the equation x = a^t, we deduce that t = log x where the log is to base a, but this involves a much later way of thinking. Here we are really thinking of log as a function, while early workers in logarithms thought purely of the log as a number which aided calculation. It may have been Jacob Bernoulli who first understood the way that the log function is the inverse of the exponential function. On the other hand the first person to make the connection between logarithms and exponents may well have been James Gregory. In 1684 he certainly recognised the connection between logarithms and exponents, but he may not have been the first.

As far as we know the first time the number e appears in its own right is in 1690. In that year Leibniz wrote a letter to Huygens and in this he used the notation b for what we now call e. At last the number e had a name (even if not its present one) and it was recognised. Now the reader might ask, not unreasonably, why we have not started our article on the history of e at the point where it makes its first appearance. The reason is that although the work we have described previously never quite managed to identify e, once the number was identified then it was slowly realised that this earlier work is relevant. Retrospectively, the early developments on the logarithm became part of an understanding of the number e.

We mentioned above the problems arising from the fact that log was not thought of as a function. It would be fair to say that Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialum seu percurrentium. The work involves the calculation of various exponential series and many results are achieved with term by term integration.

So much of our mathematical notation is due to Euler that it will come as no surprise to find that the notation e for this number is due to him. The claim which has sometimes been made, however, that Euler used the letter e because it was the first letter of his name is ridiculous. It is probably not even the case that the e comes from "exponential", but it may have just be the next vowel after "a" and Euler was already using the notation "a" in his work. Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e. He showed that

e = 1 + ¹/_1! + ¹/_2! + ¹/_3! + ...

and that e is the limit of (1 + ¹/_n)ⁿ as n tends to infinity. Euler gave an approximation for e to 18 decimal places,

e = 2.718281828459045235

without saying where this came from. It is likely that he calculated the value himself, but if so there is no indication of how this was done. In fact taking about 20 terms of 1 + ¹/_1! + ¹/_2! + ¹/_3! + ... will give the accuracy which Euler gave. Among other interesting results in this work is the connection between the sine and cosine functions and the complex exponential function, which Euler deduced using De Moivre's formula.

Interestingly Euler also gave the continued fraction expansion of e and noted a pattern in the expansion. In particular he gave

and

Euler did not give a proof that the patterns he spotted continue (which they do) but he knew that if such a proof were given it would prove that e is irrational. For, if the continued fraction for (e - 1)/2 were to follow the pattern shown in the first few terms, 6, 10, 14, 18, 22, 26, ... (add 4 each time) then it will never terminate so (e - 1)/2 (and so e) cannot be rational. One could certainly see this as the first attempt to prove that e is not rational.

The same passion that drove people to calculate to more and more decimal places of p never seemed to take hold in quite the same way for e. There were those who did calculate its decimal expansion, however, and the first to give e to a large number of decimal places was Shanks in 1854. It is worth noting that Shanks was an even more enthusiastic calculator of the decimal expansion of p. Glaisher showed that the first 137 places of Shanks calculations for e were correct but found an error which, after correction by Shanks, gave e to 205 places. In fact one needs about 120 terms of 1 + ¹/_1! + ¹/_2! + ¹/_3! + ... to obtain e correct to 200 places.

In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula i^-i = (e^p). In his lectures he would say to his students:-

Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.

Most people accept Euler as the first to prove that e is irrational. Certainly it was Hermite who proved that e is not an algebraic number in 1873. It is still an open question whether e^e is algebraic, although of course all that is lacking is a proof - no mathematician would seriously believe that e^e is algebraic! As far as we are aware, the closest that mathematicians have come to proving this is a recent result that at least one of e^e and e to the power e² is transcendental.

Further calculations of decimal expansions followed. In 1884 Boorman calculated e to 346 places and found that his calculation agreed with that of Shanks as far as place 187 but then became different. In 1887 Adams calculated the logarithm of e to the base 10 to 272 places.