WEEKLY WHINE

The first few perfect numbers

Pythagoras [c 569 BC-c 475 BC] was the first mathematician known to have studied perfect numbers. Chances are, though, that perfect numbers have been studied as long as mathematics itself.

Mathematicians once worked with a number's aliquot parts rather than its divisors. The phrase "aliquot part" is just secret code for proper divisors. A number's proper divisors are its factors excluding itself: the aliquot parts of 6 are 1, 2, and 3.

Note that 1 + 2 + 3 = 6. Six just happens to be the first perfect number, defined as those such that the sum of its aliquot parts is equal to itself. Since early times, four perfect numbers were known; we haven't any 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

Euclid [c 325 BC-c 265 BC], in his celebrated Elements [c 300 BC], wrote that if you sum a series of powers of two until you get a prime, and then multiply that prime by the last power of two, you'll get a perfect.

Try it: 1 + 2 + 4 = 7, and 7 · 4 = 28. 1 + 2 + 4 + 8 + 16 = 31, and 31 · 16 = 496. Of course, knowing that the sum of the first n powers of two gives 2^{n + 1} - 1, we can rewrite Euclid's proposition as follows:

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

Nicomachus [c AD 60-AD 120] made the next major advancement with respect to perfect numbers. In his Introductio Arithmetica [c 100], he stated five assertions without proof:

1. Perfect number n has n digits.
2. Every perfect number is even.
3. Perfect numbers alternate between ending in 6 and 8.
4. Euclid's method [above] gives all perfect numbers.
5. There are infinitely many perfect numbers.

Nicomachus didn't prove these but did show that the four known perfects satisfy them. Indeed, for many years, they were taken as true without proof.

After the Greeks, the next mathematicians to be fascinated by perfect numbers were the Arabs. One Arab mathematician, Ismail ibn Ibrahim ibn Fallus [1194-1239], wrote a treatise based upon Nicomachus's work. He also included a list of ten numbers that he claimed were perfect, starting with the original four. Though the last three turned out incorrect, the first seven members of his list are now known to be the first seven perfect numbers.

As the Renaissance finally hit Europe at the start of the 16th century, nothing more had been added to the store of knowledge about perfect numbers, not even ibn Fallus's results. In his 1536 work Utriusque Arithmetices, Hudalrichius Regius showed that 2¹¹ - 1 = 2047 = 23 · 89. This was the first prime number p found such that 2^p - 1 is composite. At the same time, he showed that 2¹³ - 1 = 8191 is prime. This made 2¹²(2¹³ - 1) = 33,550,336 the fifth perfect number.

With this, Nicomachus's first assertion was disproved: there are no five, six, or seven digit perfects.

J Scheybl found the sixth perfect number in 1555 and mentioned it in commentary for a translation of Euclid's Elements. Unfortunately, nobody noticed until 1977.

Pietro Antonio Cataldi [1548-1626] in 1603 factored all numbers up to 750, giving a list of 132 prime numbers within that range. With this list, he could determine whether any number less than 750² = 562,500 is prime simply by checking for prime divisors below that number's square root. He first showed that 2¹⁷ - 1 = 131,071 is prime, making the sixth perfect number 2¹⁶(2¹⁷ - 1) = 8,589,869,056. Nicomachus's third assertion was thus disproved. For an encore, Cataldi found that 2¹⁹ - 1 = 524,287 is prime. He thereby succeeded in replicating two of ibn Fallus's three discoveries: 2¹⁸(2¹⁹ - 1) = 137,438,691,328 is the seventh perfect number.

Cataldi once wrote of discovering four more perfect numbers. He claimed that 2^{k - 1}(2^k - 1) would be perfect for

k = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37

and nothing else within that range. The first seven were already known, but only one of the last four is actually perfect.

Pierre de Fermat [1601-1665] disproved two by means of his Little Theorem, stating that for any prime p and any number n that isn't a multiple of p, a^{p - 1} - 1 is divisible by p. This theorem brought him to the conclusion that 2²³ - 1 = 8,388,607 = 47 · 178,481 and that 2³⁷ - 1 = 137,438,953,471 = 223 · 616,318,177. These two were eliminated from contention for perfect numberhood.

In 1644, Marin Mersenne [1588-1648] published Cogitata physica mathematica, which included the contention that perfect numbers come from

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

and for no other integers within that range. Of course, he admitted not being able to test all of these. Nonetheless, for the 47 prime numbers within this range, he correctly classified 42 of them. We now use the term Mersenne primes for prime numbers that can be expressed as 2^p - 1.

After more than a century, the eighth perfect number was discovered in 1732 by Leonhard Euler [1707-1783]. It confirmed both Cataldi's and Mersenne's conclusions: 2³⁰(2³¹ - 1) = 2,305,843,008,139,952,128 is a perfect. In 1738, Euler factorised 2²⁹ - 1, disproving the third of Cataldi's incorrect claims.

Euler also proved that all even perfect numbers are of Euclid's form. This in turn shows that no perfects end in 0, 2, or 4. He too erroneously claimed perfects, saying that k = 41 and k = 47 give perfects. Unlike most others, though, he discovered his errors in 1753.

After another long downtime, Edouard Lucas [1842-1891] used a new method to further test Mersenne's results. He showed in 1876 that 2⁶⁷ - 1 is composite despite being unable to find its factors. He confirmed that 2¹²⁷ - 1 is prime, yielding the perfect number 2¹²⁶(2¹²⁷ - 1) = 14,474,011,154,664,524, 427,946,373,126,085,988,481,573,677,491, 474,835,889,066,354,349,131,199,152,128.

It has been conjectured that for any Mersenne prime M, 2^M - 1 is also a Mersenne prime. This of course would verify Nicomachus's fifth assertion.

In the 1880s, Pervusin and Seelhoff independently determined that 2⁶⁰(2⁶¹ - 1) = 2,658,455, 991,569,831,744,654,692,615,953,842,176 is the ninth perfect number.

Frank Cole in 1903 presented a paper, "On the Factorization of Large Numbers", at a meeting of the American Mathematical Society. In his talk, he wrote

2⁶⁷ - 1 = 147,573,952,589,676,412,927

761,838,257,287
193,707,721

on the chalkboard. Then, he multiplied the second and third numbers to get 147,573,952,589,676,412,927. He returned to his seat with applause from the gallery - having uttered not a single word.

Powers in 1911 determined that 2⁸⁸(2⁸⁹ - 1) = 191,561,942,608,236,107,294,793, 378,084,303,638,130,997,321,548,169,216 is the tenth perfect number. A bit later, he found that 2¹⁰⁰(2¹⁰¹ - 1) = 13,164, 036,458,569,648,337,239,753,460,458,722, 910,223,472,318,386,943,117,783,728,128 is the eleventh. This makes 2¹²⁶(2¹²⁷ - 1) the twelfth.

There are now 37 known perfect numbers, all of Euclid's form. Since Powers, all were discovered with the aid of computational tools, most recently computers. The largest known prime corresponds with the largest known Mersenne prime, 2^3,021,377 - 1. So 2^3,021,376(2^3,021,377 - 1) is a perfect number. It contains 1,819,050 digits, so I decided you could do without its decimal representation. In fact, many possible Mersenne primes below this have not been eliminated, so this may not actually be the 37th perfect number.

Since all known perfect numbers are based upon Euclid's formula, we have yet to find an odd perfect number. It is known, however, that if there are any, they need at least 29 prime factors, at least eight of which are distinct, and at least one of which is greater than 1,000,000. We also have determined that there are no odd perfects less than 10³⁰⁰.

It remains to be seen whether there are any odd perfects and whether the list of perfects ever terminates.

For details on this and other stories, head for the great perfect number article on the History of Mathematics at the University of St Andrews, Scotland.