The Beauty of Marginal Notes

Marginal notes add further information elegantly and unobtrusively. We can add them with the shortcode mnote.

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.

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 out some flaws in his first preliminary version. Wiley finally published a completely convincing version in 1995 — a breakthrough in number theory that won him a few prizes.

Wiley’s proof is still only accessible to mathematical specialists. But the initial mathematical problem — Diophantus’ successful method to split a second power into the sum of two second powers and Fermat’s conjecture about the impossibility of doing the same for higher powers — are relatively easy to grasp:

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 = ¹⁶⁄₅

Thus one number is ²⁵⁶⁄₂₅ and the other ¹⁴⁴⁄₂₅. The sum of these numbers is 16 and each summand is a square.

Like to use marginal notes? Have a look at the shortcode mnote.