Marginal notes allow to add further information in an elegant and unobtrusive way. We can glance over them and stay with the main text or zone out for a while, when they pique our curiosity.

The famous mathematician Fermat wrote his last conjecture around 1637 in the marginal column besides an ancient Greek proof by Diophantus. The “marvelous proof” Fermat mentioned there has never been uncovered and his note kept mathematicians wondering for over 350 years.

Portrait of Sir Andrew John Wiles
Sir Andrew John Wiles© C. J. Mozzochi, Princeton N.J

A proof has been considered generally inaccessible until Sir Andrew John Wiles announced to have found one in 1993 after years of ground-breaking work. A few mathematicians were able to follow him and pointed some flaws out, that his first promising preliminary version did still contain. Wiley finally published a completely convincing version in 1995 — a break-through in number theory which won him a few prizes.

The proof is so advanced, that its still inaccessible to almost everyone. But the ancient proof by Diophantus’ and Fermat’s conjecture are relatively easy to follow:

To divide a given square into a sum of two squares.

To divide 16 into a sum of two squares.

Let the first summand be x2, and thus the second 16 – x2. The latter is to be a square. I form the square of the difference of an arbitrary multiple of x diminished by the root of 16, that is, diminished by 4. I form, for example, the square of 2x – 4. It is 4x2 + 16 – 16x. I put this expression equal to 16 – x2. I add to both sides x2 + 16x and subtract 16. In this way I obtain 5x2 = 16x, hence x = 165

Thus one number is 25625 and the other 14425. The sum of these numbers is 16 and each summand is a square.

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