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Fermat's Last Theorem: Definition & Example

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  • 0:03 Fermat's Last Theorem
  • 1:26 The Pythagorean Theorem
  • 2:49 Simple Yet Complex
  • 3:19 How It Works
  • 3:40 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Considered a mathematical mystery for hundreds of years, watch this video lesson to learn what Fermat's Last Theorem tells us and why mathematicians struggled for years to prove it true.

Fermat's Last Theorem

Fermat's Last Theorem is a theorem which Pierre de Fermat wrote down in the margins of a book he had back in the 1600s. It is called his last theorem because this writing was discovered some 30 years after he had died. What he wrote was that he had proved how you can't have the sum of two positive numbers taken to a power greater than 2 equal a third positive number taken to that same power. In math form, we would say that the equation a^n + b^n = c^n cannot be true for n > 2.

Why should you care about this information? It apparently doesn't give you any useful formulas or equations to use since it only tells you what is not true. You should care because it is these kinds of problems that drive mathematicians to keep thinking and discovering new ways of thought, which allow for better calculations.

Even though Fermat had claimed he had proven it, no proof has ever been found. It wasn't until the 1990s that a successful proof was published. The proof for Fermat's Last Theorem had been an unsolved problem for 300 years until it was solved in the 1990s. For mathematicians, the process of proving a statement such as this provides a way to come up with new, useful math methods.

The Pythagorean Theorem

Now that we've talked about why this is important, let's go back to Fermat's Last Theorem. We know that the letters a, b, and c stand for positive numbers and n is the exponent to which all the letters are raised. According to Fermat, this equation is not true when n > 2. So, does that mean that the equation is true when n = 2?

Let's see. When we have n = 2, we have a^2 + b^2 = c^2. Hmm, that looks like an awfully familiar equation, doesn't it? Why, it's our Pythagorean Theorem that tells us that for right triangles the square of the hypotenuse equals the sum of the squares of the other two sides. We know for a fact that this is true because we have visual proof with our right triangles. Any right triangle will prove the Pythagorean Theorem true.

So, really, Fermat's Last Theorem is a kind of a generalized Pythagorean Theorem, only it is telling you that any exponent greater than 2 is not possible. Fermat might have been looking at the Pythagorean Theorem and wondered if the equation would work if the exponent is 3 or 4 or any number greater than 2. We know now that what he found was that the equation does not work for exponents greater than 2.

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