Perfect Numbers

Perfect numbers were first (to our knowledge) investigated by the Pythagoreans about 2500 years ago. They felt a deep reverence for such numbers, and in fact felt that all numbers had meaning and significance far beyond their purely mathematical properties. For example, even numbers were female, odd numbers male; 3 was the number of marriage (sum of first male and first female numbers); 4 was the number of justice (we still talk about a square deal, meaning a fair one); 10 = 1+2+3+4 was revered as the tetractys, a very special number indeed. This ``number mysticism'' was a great inspiration to the Pythagoreans to study numbers diligently, and they are often regarded as the founders of Pure Mathematics, certainly of the purest branch of mathematics: Number Theory. They defined other interesting categories of numbers related to perfect numbers.

First, let us introduce the notation s(n) (``sigma n'') for the sum of all divisors of n. The Pythagoreans called a number n deficient if s(n) < 2n and abundant if s(n) > 2n. Perfect numbers have s(n) = 2n. What are the deficient and abundant numbers less than 30? For example, 1+2+4 = 7 < 8 shows that 4 is deficient; 1+2+3+4+6+12 = 28 > 24 shows that 12 is abundant.

The Pythagoreans also pondered over friendly (or amicable numbers. A pair of numbers is friendly if each is the sum of the proper divisors of the other. For example (due to the Pythagoreans), 220 and 284 are friendly, because:

1+2+4+5+10+11+20+22+44+55+110 = 284,  1+2+4+71+142 = 220.

Charms have been sold through the ages, with those two numbers engraved upon them, said to bring love to the bearer! In the Book of Genesis, Jacob presented his brother Esau (whom he had cheated out of his birthright) with 220 goats; was this perhaps a symbolic profession of his love for his brother? Did he hope Esau would give him 284 sheep in response?! Simon Singh, in his wonderful book Fermat's Last Theorem, mentions an Arabic practice of carving the numbers 220, 284 on two fruits, eating one, and offering the other to one's lover, as a form of mathematical aphrodisiac (surely no less efficacious than powdered rhino horn - and certainly more environment-friendly).

Before leaving the topic of friendly numbers, you will want to know what other amorous couples there are. The next pair of such numbers took a very long time to discover, for it requires a great familiarity with numbers, great patience, and a love of playing with them. Pierre Fermat found the pair 17 296, 18 416 in 1636, and set other mathematicians searching seriously for them. René Descartes found a third pair, 9 363 584,  9 437 056, and Leonhard Euler found an amazing sixty-two friendly pairs! One of those strange quirks of history occurs here, for somehow they had all missed a much smaller pair of friendly numbers, finally discovered by a sixteen-year-old Italian boy, Nicolò Paganini, in 1866. They are 1 184, 1 210. Can you verify his discovery?

Now, how do mathematicians (and sixteen-year-old boys) go about the task of analysing and looking for friendly, deficient, abundant and perfect numbers? By the Unique Factorization Theorem (first clearly stated and rigorously proved by the ancient Greeks and included in Euclid's Elements), each number n can be expressed uniquely as a product of its prime factors. That is, n = p₁^x₁.p₂^x₂.p₃^x₃¼p_m^x_m, where p₁ < p₂ < p₃ < ¼ < p_m are primes, and x₁,x₂,x₃¼x_m are positive integers. The divisors of n are all possible combinations of powers of these primes, taking the form n = p₁^y₁.p₂^y₂.p₃^y₃¼p_m^y_m where 0 £ y_i £ x_i for all i. The sum of all the divisors of n is therefore

s(n) = (1+p1+p12+¼p1x1)(1+p2+p22+¼p2x2)¼(1+pm+pm2+¼pmxm) = æ è p1x1+1-1 p1-1 ö ø . æ è p2x2+1-1 p2-1 ö ø . æ è p3x3+1-1 p3-1 ö ø .¼ æ è pmxm+1-1 pm-1 ö ø ;

and the total number of divisors of n is given by the product:

(x1+1)(x2+1)(x3+1)¼(xm+1).

For example, take n = 36 = 2²·3². The divisors of 36 are the terms of the product: (1+2+2²)(1+3+3²), that is: 1, 2, 2², 3, 2·3, 2²·3, 3², 2·3², 2²·3². It has (2+1)(2+1) = 9 divisors, and

s(36) = 22+1-1 2-1 · 32+1-1 3-1 = (8-1) æ è 27-1 3-1 ö ø = 7·13 = 91 > 2(36).

Hence 36 is abundant.

For any number n, these three steps have been achieved so far:

1. Computation of s(n), and hence classification of n as perfect, deficient or abundant.
2. Calculation of the number of divisors of n.
3. Listing of the divisors of n.

1. What is the next perfect number after 6? How many are there up to 100? (There are surprisingly few!)
2. Classify the numbers up to 30 as perfect, deficient or abundant.
3. Verify Nicolò Paganini's discovery that 1 184, 1 210 are a friendly pair.
4. Consider prime p. Can we say anything definite?
5. What about a power of a prime pⁿ?
6. Consider next any even number of the form 2ⁿp, where p is prime. Show that this is perfect whenever 2ⁿ⁺¹-1 = p. This shows that whenever n is such that 2ⁿ⁺¹-1 is prime, we have a perfect number 2ⁿ(2ⁿ⁺¹-1).
7. A natural question which arises: Is every even perfect number of the form 2ⁿ(2ⁿ⁺¹-1) given in question 6? Now, this question bothered Euler, and he did produce a proof that the answer is YES. We give you the idea on page , and challenge you to complete the proof, for a prize.
8. Since the problem of finding even perfect numbers is therefore reduced to finding primes of the form 2^k-1, two important questions arise immediately: What can we say about k, and what can we say about a, when a number of the form a^k-1, a ³ 1, k ³ 2 is to be prime? Show that a has to be 2 (for otherwise there's an obvious factorization!), and k has to be prime too: if k = rs can you see a factorization?
9. Can a perfect square be perfect? We are also offering this one as a challenge to readers - for a prize.

2  Mersenne Primes

(1) When will 2^k-1 be prime?

(2) How many such primes are there?

(3) Are ALL perfect numbers of the form 2^k-1(2^k-1)? That is (because of Euler's result), are ALL perfect numbers EVEN?

Father Mersenne (1588-1648) has an even more important claim on our gratitude - he was a kind of one-man early Scientific Society, acting as a clearing house for ideas in science and mathematics, corresponding with everyone who was anyone in Europe, and striving to break down the culture of secrecy that bedevilled the intellectual world of the time. It was he who urged reclusive mathematicians like Fermat to make their methods known, and it was `Monsieur Fermat's method of drawing tangents' that inspired Isaac Newton to discover the Calculus.

Unfortunately, Father Mersenne had missed out some of his Mersenne primes: M₆₁, M₈₉, M₁₀₇, which only goes to remind us how arduous were the difficulties facing people calculating with such enormous numbers before our age of electronic computers. That he made any mistakes was first discovered in the late nineteenth century by a man called Pervouchine, who showed 2⁶¹-1 to be prime. And Mersenne was also wrong about k = 67 giving a prime; but nobody knew he'd made a mistake there until 1903 when a man called Frank Cole gave a lecture which earned him a standing ovation from his audience: his lecture consisted simply of writing on the blackboard the following:

267-1 = 193,707,721×761,838,257,287.

Any one of you could check that with pencil and lots of paper - though it might take you a couple of days and challenge your powers of concentration. But to find the factorization in the first place, without the benefit of modern computer aid, was a tremendous feat. During the course of the 20th century we developed powerful machines to help us with such tasks, and you may well ask: Where has the search for Mersenne primes and perfect numbers got to, by the year 2001?

By 1947 the list of Mersenne numbers up to k = 257 was finally safely checked, and, halfway through the century, we knew for certain the first twelve Mersenne primes: for k = 2,3,5,7,13,,17,19,31,61,89,107,127, - hence twelve perfect numbers. Then, as computing power grew vastly more efficient, twenty-one more were found by 1994: the latest Mersenne primes known then were for k = 756,839, and k = 859,433. These represented also the greatest known prime numbers, and (calculated from them) the greatest known perfect numbers. Then someone found a bigger Mersenne prime: for k = 1,398,269. The race hotted up as announcements of latest greatest primes could flash across the internet and bring immense financial benefits to the manufacturers of computer hardware or software that thus proved its superiority! Computer companies began to employ mathematicians and programmers to engage in the race for the next prime! And almost all the great prime trophies discovered in the race are Mersenne primes - in spite of the fact that only a tiny proportion of Mersenne numbers are primes, it's still about the most effective way of looking for primes.

On August 24 1997, Gordon Spence, a thirty-eight-year-old Information Technology manager in England, and George Woltmann, a thirty-nine-year-old programmer in Florida USA, announced the discovery (using a Pentium computer with two weeks computing time!) that 2^2,976,221-1 is prime. That number written in ordinary decimal notation is nearly 900 thousand digits long - roughly equivalent to the first eleven books of the Bible.

In the last few years, an enlightened modern Mersenne, in the anti-secretive, anti-competitive spirit of the original Father M, has conceived a wonderful, cooperative venture called the Great Internet Mersenne Prime Search (known as GIMPS), which puts many amateur prime detectives in touch with each other and seeks to combine not only their intellectual and physical energies, but the computing power and time of their machines! Perhaps in the next issue we will tell you more about this grand co-operative adventure of minds and machines, and what is the first new prime (hence greatest known perfect number) of the new century!

Before ending this article, we should let you into an important secret: the answers to questions (2) and (3) at the beginning of this article are still UNKNOWN to any human being, and are currently being chased with all their hearts, minds and machines, by many mathematicians:

Open Question 1: Is every perfect number even?

Open Question 2: Is there a largest perfect number, or do they go on for ever?