CHAPTER 2 The End of the World     Truly the gods have not from the beginning revealed all things to mortals, but by long seeking, mortals makeprogress in discovery. —Xenophanes of Colophon The pure mathematician, like the musician, is a free creator of his world of ordered beauty. —Bertrand Russell, A History of Western Philosophy

 

 

“Where the hell is Pythagoras?”

“Sir, our electronic fly indicates he’s taking a nap beneath an old chariot in his back yard.”

You remove a piece of yellowed paper from the marble floor of the temple. On the paper is some text beneath a woodcut showing apocalyptic horsemen flying through the sky on Judgement Day. (See p. 43)

“Ah, never mind Mr. Plex, we can use the time to examine a strange article one of my agents found in the year 1947.” You pause. “There’s always been predictions of the end of the world based on numbers. But predictions usually don’t appear in serious mathematical journals.” You raise your eyebrows. “This one appeared in a January 1947 issue of the American Mathematical Monthly.”

“Sir, let me see that,” Mr. Plex says in a voice an octave too high.

Mr. Plex grabs the tattered paper from your hand and begins to read from the article:

The famous astrologer and numerologist Professor Umbugio predicts the end of the world in the year 2141. His prediction is based on profound mathematical and historical investigations. Professor Umbugio computed the value from the formula

W = 1492ⁿ – 1770ⁿ – 1863ⁿ + 2141ⁿ

for n = 0,1,2,3, and so on up to 1945, and found that all numbers which he so obtained in many months of laborious computation are divisible by 1946. Now, the numbers 1492, 1770, and 1863 represent memorable dates: the discovery of the New World, the Boston Massacre, and the Gettysburg Address. What important date may 2141 be? That of the end of the world, obviously.

Mr. Plex lowers the slightly soiled slip of paper. “Sir, this is incredible. Could all the numbers produced by the formula be divisible by 1946? Could it be that 2141 has anything to do with the End of the World?”

You reach under a statue of Zeus and toss a notebook computer to Mr. Plex. “Write a program, and see what numbers you get.”

The Apocalyptic Horsemen. Woodcut by Hans Burgkmair, from Das Neue Testament printed by Silvan Othamar, Augsburg, 1523.

Mr. Plex begins to type, and he soon hands you the results on a computer printout. The E symbols are the computer’s way of representing scientific notation. For instance, 1.00E+02 would be another way of denoting 1.00 × 10² or 100.

 

N	W
1	0
2	206276
3	1.124106E+09
4	4.106015E+12
5	1.256519E+16
6	3.478795E+19
7	9.035302E+22
8	2.246103E+26
9	5.410357E+29
10	1.272996E+33

 

“Sir, the numbers grow awfully quickly! If the units were in years, the fifth value is larger than the number of years required for all the stars to have died out.” Mr. Plex begins to pace. “How could scientists in the year 1946 determine that the results were all divisible by 1946? What is the W value for n = 100? Are the W numbers always divisible by 1946, or do they cease to have that property after n 1945?”

“Mr. Plex—all very interesting unanswered questions. But they’ll have to wait.” You point to the-view screen and see Pythagoras rubbing his eyes and stretching. “Mr. Plex, our friend is waking up.”

THE SCIENCE BEHIND THE SCIENCE FICTION

The “End of the World” formula really did appear in the following reference: Starke, E. (1947), Professor Umbigo’s Prediction, American Mathematical Monthly, January, 54:43–44. I believe that all W numbers, even ones produced for n > 1945, are divisible by 1946. A detailed mathematical proof of this can be found in the American Mathematical Monthly. The proof relies on the fact that x – y is a divisor of xⁿ – yⁿ for n = 0,1,2,.…

The Appendix contains a BASIC program listing for computing Umbugio’s End-of-the-World numbers.