## Sunday, October 4, 2015

### HOMER SIMPSON’S COUNTER EXAMPLES FOR FERMAT’S LAST THEOREM

10/04/2015

HOMER SIMPSON’S COUNTER EXAMPLE FOR FERMAT’S LAST THEOREM

Fermat’s Last Theorem states that though the Pythagorean theorem (a2 + b2 = c2) has an infinite number of solutions for a, b, and c there are no solutions if the exponents are larger than 2 (an + bn = cn, n > 2).
Fermat did not provide a proof of this theorem before his death.  Since his death in 1665 many mathematicians have tried to complete a proof of this theorem, or to find a counter example, thus disproving the theorem without success until 1994 when Andrew Wiles completed his now famous proof.
In the television show “The Simpson’s”, in an episode titled "Treehouse of Horror VI" a counterexample of the theorem appeared briefly: “178212 + 184112 = 192212”.  Another counter example appeared in the episode "The Wizard of Evergreen Terrace" (1998): 3,98712 + 4,36512 = 4,47212.
Both of these “counter examples” are false, but they did fool many people.  How could this happen?  Let’s look at the second equation:
3,98712 + 4,36512 = 4,47212.
The left side of the equation equals: 63,976,656,349,698,612,616,236,230,953,154,487,896,987,106.
The right side of the equation equals: 63,976,656,348,486,725,806,862,358,322,168,575,784,124,416.
The two sides of the equation are not equal.  In fact the left side of the equation is 1,211,886,809,373,872,630,985,912,112,862,690 larger than the right side of the equation.  How could we miss this large of a difference?  Actually, it’s not too hard to do.
Notice that the first 10 digits of the numbers representing value of each side of the equation are identical.  A calculator that just shows eight digits cannot even tell the difference between them.
If you calculate the ratio of the two sides (but don’t use one of those eight digit calculators) you will find that there is only about 2 billionths of a percent difference between the two numbers.  See – it helps to be able to calculate with really big numbers.  And really big numbers can be very interesting.
(398712 + 436512) / 447212 = 1.0000000000189426406214887790...
If you would like to know more about Pierre Fermat and the Simpson’s you should visit the YouTube Numberphile Channel.  Simon Singh appears in the videos listed below, and does a superb job filling you in on both the mathematics involved and the Simpson’s/
I also highly recommend Simon Singh’s book “The Simpsons and Their Mathematical Secrets”.  Apparently there are several math nerds on the staff of “The Simpson’s” – and they have a pretty good sense of humor too.

David