Higher Math from The Simpsons TV Show

Higher Math from The Simpsons TV Show: Fermat's last theorem

(A long post about higher math which young people can understand. )

The New York Times recently reviewed a book about The Simpsons TV show and higher math. Occasionally, Bart Simpson writes higher math on the blackboard. This is a surprise because Bart Simpson is supposed to be a dunce who hates math. In several episodes, Bart writes equations that seem to refute Fermat's last theorem.

For more on this, Google Fermat Simpsons.

Fermat's last theorem states that x^n + y^n = z^n has no solutions when x, y, and z are integers and n is greater than 2. x^n means "x to the n power" for example x^2 means "x squared."

x squared + y squared = z squared has an infinite number of solutions, for example, x = 6, y = 8, z = 10. These solutions are called "Pythagorean Triples" and there are an infinite number of them.

It is remarkable that (x squared + y squared = z squared) has an infinite number of integer solutions but

x cubed + y cubed  = z cubed has no integer solutions.

One of Bart's "equations" is 1782^12 + 1841^12 = 1922^12.

Try this on a scientific calculator. For example, on the TI-30XS Multiview, the key to raise to a power is a key like this: [^] -- it is the middle key on the left column of keys.

You would key in: 1782[^]12 [+] 1841[^]12 [enter], you will get an 8-digit number in scientific notation, ending in 10 to the 39th power. Then key in 1922[^]12 and you will also get a number in scientific notation. The two numbers appear to be equal, but according to Fermat's last theorem, they cannot be.

If you don't have a scientific calculator, you can get one by doing a search in Google for scientific calculator and one will appear. The key to raise a number to a power is marked [x to the y power]; you would put in for example 1922 [x to the y power] 12 [=].

It is easy to show that the equation above is false. 1782 is even, and 1782 raised to the 12th power is also even; it is 1782 multiplied by itself 12 times. But 1841 is odd, and so is 1841 raised to the 12th power. But even + odd is odd, and 1922 raised to the 12th power is even. So we apparently have odd = even, which is impossible.

To see what is really going on, click on the purple  link to big number scientific calculator . Put in the left side of the equation; you will get a long number, approximately 40 digits long. Put in the right side of the equation; you will again get a long number, approximately 40 digits long. Use the "history" key to see the two numbers together. At least the first 8 digits will be the same, but at some point, the digits of the two numbers will not be the same.

Another Bart Simpson "equation" is:

3987^12 + 4365^12 = 4472^12

This is also false. The numbers 3987 and 4365 are both divisible by 9: the sum of their digits is divisible by 3.

But 4472 is not divisible by 9. If the equation were true, if one side were divisible by 9, both sides would be divisible by 9.

You can go through the same routine with the "equation" above. Evaluate both sides with a scientific calculator. You will get two numbers that end with 10 to the 43rd power and appear to be equal. If you  don't have a scientific calculator, Google scientific calculator to get one. Go to the purple link

link to big number scientific calculator enter the left side and the right side and you will get two 44 digit numbers, use "history" key to see both numbers, you will find that the first 10 digits agree but the 11th digit is 8 in one number and 9 in the other.

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About Fermat's Last Theorem and "Pure Mathematics"

(For more about this, Google Pierre Fermat)

Pierre de Fermat was a French lawyer and amateur mathematician who lived from 1601 to 1665. That is a long time ago. For example, The Pilgrims landed on Plymouth Rock in 1620. Fermat wrote his "conjecture" in the margin of a book about 1637, claiming to have a "marvelous proof" that would not fit in the margin.

All of Fermat's writings, including scribbles in the margins of books, were collected and printed as a book by his son. Fermat's theorem is "last" because it was the last to be proven. For hundreds of years, mathematicians searched for a proof. At last, it was proven in 1995, about 358 years after Fermat first wrote it.

Meanwhile, the Fermat problem gave rise to a branch of mathematics called Number Theory. Number Theory is so called "pure mathematics" because it had no apparent applications when it started. In fact, many applications have been found. But, if you want a sample of "pure mathematics" Fermat's Last Theorem, Bart Simpson's "equations" and the simple proofs that they are wrong, and even the use of "big number" calculators will give you a taste of Number Theory. 

Number Theory courses at the University of Minnesota have numbers in the 5000 range, meaning that they are intended for graduate students majoring in Mathematics.