Jul
29
More fun with numbers
Filed Under Sunrise on KGMB9
I’m rereading “Fermat’s Enigma,” about the solving of Fermat’s Last Theorem, and not for the first time I’m struck by how much fun numbers are, and how none of my math teachers was able to convey this information to me when I could have used it.
Different numbers have different personalities. Odd numbers behave differently from even numbers. Some numbers are irregular and others are even irrational. And some are SO negative!
I did not know, until I read this book, about “friendly numbers.” There are several pairs of numbers in which the factors of one (the numbers by which they can be divided with no remainder) add up to the other number, and vice versa.
I learned something cool about “pi,” the ratio of a circle’s circumference to its diameter. It starts as 3.14159 but continues into infinity with, unlike many other infinitely fractional numbers, no perceptible pattern of repetition. But that’s really only true when expressed as numbers past a decimal point. You can also express it as 4 x {1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17…} and so forth into infinity. So there is a pattern after all, and an elegant one. Cool, huh?
And Fermat’s Last Theorem? That has been considered for centuries the toughest mathematics problem of all time. The easiest way to explain it is to start with something we all learned in school, the Pythagorian Theorem: for any right-angle triangle, the square of the two sides that meet at a right angle — “a” squared plus “b” squared — equals the square of the third side, “c” squared.
Fermat was an amateur mathematician in France who wrote in the margin of a book that he had figured out that a-squared plus b-squared may equal c-squared, but if you cube the values instead of squaring them, it doesn’t work, and indeed it doesn’t work for any factor higher than that. “I have found the most wonderful proof of this,” he wrote, but in Latin, “which this margin unfortunately is too small to contain.”
Fermat wrote that in 1637. It was discovered after his death. And for three and a half centuries the greatest minds tried and failed to work up a proof for Fermat’s conjecture.
Words are my business, so I can’t resist pointing out a theorem is only a conjecture until it’s proven, and until someone else could prove that Fermat was right, it was only conjecture that he had truly found a theorem. But I don’t mind the “last” in “Fermat’s Last Theorem,” because though it was not the last thing he came up with, it was indubitably the last one to be figured out by those who followed him.
The problem is all the more amazing when you hear that Fermat did actually write down a proof that the Pythagorean Theorem doesn’t work if you switch from the power of 2 to the power of 4, and Leonhard Euler proved it didn’t work for the power of 3. What won’t work for the powers of 3 or 4 won’t work for multiples of 3 or 4, either, so, really, much progress was made a long time ago.
Mathematician Andrew Wiles announced his proof in 1993, then found a gap in his logic, and published the proof only in 1995 after he and a colleague figured out how to patch the problem. The actual proof is so complicated that there is a new conjecture that Fermat didn’t actually find a proof, only thought he had.
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