Last Thursday I posted a bunch of links, and included:
I can't remember now why I did this, but I'm still intrigued... I went to Wolfram Alpha and typed: factor 1782^12+1841^12. It's just a bunch of big numbers. Why do I like it?On Saturday night Joshua Zucker replied:
why 1782^12+1841^12? I don't know why to factor it, but of course it equals 1922^12 (just try it on your calculator, not at Walpha of course!)Well, I wasn't thinking of the consequences, and I believed him. I was impressed that he could use his calculator to factor the huge number you'd get from 178212+184112. I didn't reach for my calculator, being comfortably ensconced in my recliner. No, I reached for Google, and I googled Josh, because I was curious about what math he might lead me to.
I found a comment he made on Cut the Knot many years ago (in 2000). So I started exploring Cut the Knot, and thoroughly enjoyed some discussions about how math uses words differently from their common usage.
Eventually I remembered my original quest and googled "1782^12+1841^12". The very first thing I got was:
Fermat's last theorem. Statement that there are no natural numbers x, y, and z such that x^n + y^n = z^n, in which n is a natural number greater than 2. ...[Fermat's last theorem was proved in 1995 by Andrew Wiles. There are lots of whole number solutions to x2 + y2 = z2, like 32 + 42 = 52. But the theorem says there are no solutions to x3 + y3 = z3, nor to equations like that with higher powers.]
Huh? But Josh said 178212+184112=192212?? Next entry I clicked on was about a Simpson's episode:
In the 1995 Halloween episode of the award-winning animated sitcom The Simpsons, two-dimensional Homer Simpson accidentally jumps into the third dimension. During his journey in this strange world, geometric solids and mathematical formulas float through the air, including an innocent-looking equation: 178212 + 184112 = 192212. Most viewers surely ignored this bit of mathematical gobbledygook.Ahh, now I get it! And I finally had a vague memory of reading a post somewhere about how 178212 + 184112 and 192212 look exactly the same if you evaluate them on a calculator. (Try it!) That must be why I had originally gone to Wolfram Alpha with this.
On the fan discussion site alt.tv.simpsons, however, the equation caused a bit of a stir. “What’s going on, he seems to have disproved Fermat’s last theorem!” one fan marveled, referring to the famous claim by Pierre de Fermat—proved just months earlier—that for any exponent n bigger than 2, there are no nonzero whole numbers a, b, and c for which a^n + b^n = c^n. The Simpsons equation, if correct, would be a counterexample to the theorem, meaning that the proof had been wrong.
Meanwhile, here's the other treasures I found: