In his copy of Diophantus’s
Arithmetica, Pierre de
Fermat famously wrote:
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.
Or, in English:
It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
The first proof of that conjecture was published by
Andrew Wiles in 1995. Given the mathematical “technology” it requires, Fermat’s proof (if valid) would presumably have been much shorter.
But Fermat’s penultimate theorem was that the margin of that book was too narrow to contain his proof. Although that was probably true at the time of that writing, advances in printing technology now permit such printing.
An image of the edition of Diophantus’s
Arithmetica in which Fermat wrote his conjecture shows that the page’s size is approximately 20.3 cm x 30.5 cm, a total area of 619 cm
2. Of this, approximately 369 cm
2 contain print, leaving approximately 250 cm
2 of margin space.
Wiles’s proof in
two parts was a total of 129 pages in length. (Of this, some is not necessary to the proof itself: a couple of images of Fermat and Wiles, a bit of biographical stuff, the large-fonted title, some blank areas. If we take this factor into account, the total length of the proof might be reduced by a couple of pages.)
In that document, the printed area of each page is approximately 14.5 cm x 23.9 cm, an area of 346.6 cm
2. This gives a total printed area of about 44,700 cm
2. Thus to print that proof in the available margin space in the original book, the proof must be reduced in area by a factor of (44700/250) or 178.8. This corresponds to a linear scaling factor of √178.8 or 13.4 .
Wiles’s proof was, for the most part, printed in 12 pt, using fonts common to TeX. So if it were instead printed in 0.9 pt (with diagrams, blocks of mathematical operations, etc. reduced proportionately), it would fit in the given area. Legible printing at 0.9 pt isn’t within the abilities of standard consumer printers, as of this writing, but is easily handled by high-end equipment.
QED.
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