# Frank Morgan's Math Chat

February 18, 1999

### A ONE-PAGE PROOF OF FERMAT?

MATH JOKES. Math Chat has received a number of mathematical jokes, most of them terrible. This week's winner is a riddle from Peter Hegarty:

RIDDLE. Which mathematical term is named after a well-known American politician?

The answer is near the end of this column.

OLD CHALLENGE (Joe Shipman). Select the best occurrence in the world of each number from 1 to 10. For example, 12 is the number of eggs in a dozen or the number of months in a year.

ANSWER. Here is the winning response compiled from John Robertson, Henry Ricardo, Bob Swanson, Jean-Pierre Carmichael, and Ryan Grove:

1. Number of Earth's moons
2. Romeo and Juliet
3. Dimensions; Mazur, Ribet, and Wiles (contributors to the proof of Fermat's Last Theorem)
4. The Four Freedoms (FDR, 1941: freedom of speech, freedom of worship, freedom from want, freedom from fear)
5. Platonic solids [cube, octahedron, tetrahedron, dodecahedron, icosahedron]
6. Legs on insects
7. Notes of the musical scale
8. Cylinders in a V8 engine (such as in the Ferrari 308 series); vegetables in V8 juice (tomatoes, carrots, celery, beets, parsley, lettuce, watercress, and spinach)
9. Planets; Beethoven symphonies
10. Fingers

(Can readers improve on this list?) Then there is this from Al Zimmermann:

1. The amount, in cents, of the recent postage increase for first class US mail.
2. The number of sides to every story.
3. The number of moving parts in a Wankel engine.
4. The number of Beatles.
5. The number of players on a basketball team.
6. The number of sodas in a six-pack.
7. The number of days in a week.
8. The number of days in a week, according to the Beatles.
9. The number of months in a pregnancy.
10. The number of years it seems the Clinton/Lewinsky scandal has been going on.

Finally, in his beautiful response, David Shay relates that, "At the end of the Seder night, which begins the Jewish holiday of Passover, it is common to sing a song named 'Who Knows One.' This song gives an exact Jewish answer to your challenge, in the range of 1 to 13. Here it is:

Thirteen are the attributes of God;
Twelve are the tribes of Israel;
Eleven were the stars in Joseph's dream;
Ten commandments were given on Sinai;
Nine is the number of the holidays;
Eight are the days to the service of the covenant;
Seven days there are in a week;
Six sections the Mishnah has;
Five books there are in the Torah;
Four is the number of the matriarchs;
Three is the number of the patriarchs;
Two are the tables of the covenant;
One is our God in heaven and earth."

NEW CHALLENGE. Critique the following short proof of Fermat's Last Theorem sent in by reader Rob Connelly. (In perhaps the biggest mathematics news of the century, Andrew Wiles recently came up with a very long and complicated proof to this 350-year old problem.)

Fermat's Last Theorem. The equation

(1) xn + yn = zn has no positive integer solutions for n > 2.

Proposed proof. Suppose there were such a solution. Since x y, we may suppose x = y + a, z = y + b, with b > a positive integers. Consider the integer N defined by

(2) zn -1 = xn -1 + yn -1 + N.

Then

xn + yn = zn = z(xn -1 + yn -1 + N).

Solving for N yields:

N = [(y+a) n -1 (a-b) + yn -1 (-b)]/(y+b) = [F(y)]/(y+b) ,

so y+b divides F(y) and

0 = F(-b) = (a-b) n + (-b) n

0 = (b-a) n + bn > 0,