Alex's Adventures in Numberland - Alex Bellos [111]
1 + 2 = 3. 3 is prime, so, we multiply 3 with the highest double, which is 2. 3 × 2 = 6, and 6 is a perfect number.
1 + 2 + 4 = 7. Again, 7 is prime. So we multiply 7 by 4 to get another perfect number: 28.
1 + 2 + 4 + 8 = 15. This is not prime. No perfect numbers here.
1 + 2 + 4 + 8 + 16 = 31. This is prime, and 31×16 = 496, which is perfect.
1 + 2 + 4 + 8 + 16 + 32 = 63. This is not prime.
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127. This is prime and 127×64 = 8, which is perfect.
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255. This is not prime.
Euclid’s proof was, of course, done through geometry. He did not write it out in terms of numbers, instead using line segments. If he’d had the luxury of modern algebraic notation, he would have noticed that he could express the sum of doubles 1 + 2 + 4 +…as the sum of powers of two, 20 + 21 + 22 +…(Any number to the power 0 is always 1, by convention, and any number to the power 1 is itself.) It then becomes clear that any sum of doubles is equal to the next-largest double minus 1. For example:
1 + 2 = 3 = 4 – 1
or
20 + 21 = 22 – 1
1 + 2 + 4 = 7 = 8 – 1
or
20 + 21 + 22 = 23 – 1
This can be generalized according to the formula: 20 + 21 + 22 +…+ 2n–1 = 2n – 1, in other words that the sum of the first n terms of the doubling sequence starting at 1 is equal to 2n – 1.
So, using Euclid’s original declaration that ‘whenever the sum of doubles is a prime number, the product of the sum multiplied by the highest double is a perfect number’, and adding modern algebraic notation, we can arrive at the much more concise statement:
Whenever 2n – 1 is prime, then (2n – 1)×2n–1is a perfect number.
For civilizations that prized perfect numbers, Euclid’s proof was terrific news. If perfect numbers could be generated whenever 2n – 1 was prime, all that needed to be done in order to find new perfect numbers was to find new primes that were written 2n – 1. The hunt for perfect numbers was reduced to the hunt for a certain type of prime.
Mathematical interest in prime numbers written 2n – 1 might have originated because of their link to perfect numbers, but by the seventeenth century the primes had became objects of fascination in their own right. In the same way that some mathematicians were obsessed with finding pi to more and more decimal places, others were preoccupied with finding higher and higher primes. The activities are similar but opposite: whereas finding digits in pi is about trying to see smaller and smaller objects, pursuing primes is about reaching towards the sky. They are missions that have been undertaken as much for the romance of the journey, as for the possible uses of the numbers discovered along the way.
In the quest for primes, the ‘2n – 1’ generating method took on a life of its own. It wan’t going to produce primes for every value of n, but for the low numbers the success rate was pretty good. As we saw above, when n = 2, 3, 5 and 7 then 2n – 1 is prime.
The mathematician most fixated on using 2n – 1 to generate primes was the French friar Marin Mersenne. In 1644 he made the sweeping claim that he knew all the values of n up to 257 such that 2n – 1 is prime. He claimed they were:
(A109461) 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257
Mersenne was an able mathematician, but his list was largely based on guesswork. The number 2257 – 1 is 78 digits long, far too big for the human mind to check whether it is prime or not. Mersenne realized that his numbers were stabs in the dark. He said of the list: ‘all time would not suffice to determine whether they are prime’.
Time did suffice, though, as it often does in the case of maths. In 1876, two and a half centuries after Mersenne wrote his list, the French number theorist Edouard Lucas devised a method that was able to check whether numbers written 2n – 1 are prime,