Exhausting Euler
This past April 15th, we celebrated the birthday of Leonhard Euler (17071783), perhaps the greatest person who ever breathed mathematics.
But, I will share a quick story to show that even great minds like Eulers can make mistakes.
Euler tried to proved Fermat's Conjecture: There are no positive integral solutions to the equation x^{n}+y^{n}=z^{n} for n greater than 2.
Though Euler failed to produce a valid proof of Fermat's Conjecture, he was an extremely strong believer in patterns.
For example, he argued that if the special case x^{3}+y^{3}=z^{3} had no integral solutions, then neither did x^{4}+y^{4}+v^{4}=z^{4} nor x^{5}+y^{5}+v^{5}+w^{5}=z^{5}!
And then, more than 250 years later, someone produced a counterexample: 27^{5}+84^{5}+110^{5}+133^{5} = 144^{5}.
Check it out on your TI calculator!
While Euler would perhaps not be bothered by this error on his part, I am left with one thought: The computer/ calculator age has changed how we do mathematics via exhaustion!
A side note: In the early 1970's, on hearing the announcement of this counterexample, I tried to check it out on the "new" microcomputer IMSAI. I used this BASIC program...
FOR X=1 TO 200
FOR Y=1 TO 200
FOR V=1 TO 200
FOR W=1 TO 200
FOR Z=1 TO 200
IF X^5+Y^5+V^5+W^5 = Z^5 THEN PRINT "BINGO"
NEXT Z,W,V,Y,X
END
After about 4 days of constant running, the program ended and never printed out BINGO. Any idea why?
