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Exhausting Euler

This past April 15th, we celebrated the birthday of Leonhard Euler (1707-1783), 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 xn+yn=zn 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 x3+y3=z3 had no integral solutions, then neither did x4+y4+v4=z4 nor x5+y5+v5+w5=z5!

And then, more than 250 years later, someone produced a counterexample: 275+845+1105+1335 = 1445.

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"

After about 4 days of constant running, the program ended and never printed out BINGO. Any idea why?