Number Theory Archives - Page 4 of 6

Understanding the Fermat’s Last Theorem

January 16, 2011 GB College Mathematics, High School Mathematics, Number Theory

The Fermat’s Last Theorem is one of the hardest problems in the history of mathematics. The problem was written by Pierre de Fermat in 1637, and it was only solved more than 300 years later — in 1995 by Professor Andrew Wiles.

But what is exactly the Fermat’s Last Theorem?

The Fermat’s Last Theorem is an extension of the Pythagorean Theorem. Recall that the Pythagorean Theorem states that given a right triangle whose side lengths are $x, y$ and hypotenuse $z$ , $x^2 + y^2 = z^2$ is satisfied. For example, a right triangle with side lengths $2$ , and $3$ has hypotenuse $\sqrt{13}$ .

There are some interesting things that we can see if we examine the side lengths of right triangles. For instance, if we let the triples $(x,y,z)$ be the side lengths of a right triangle, where $z$ is the hypotenuse, we can find triples such that all lengths are integers. The triples $(3, 4, 5)$ , $(5,12,13)$ , $(8,15,17)$ are integer triples, and they satisfy the Pythagorean Theorem. These triples are called Pythagorean Triples. It is not also difficult to see that there are infinitely many Pythagorean Triples (Can you see why?). » Read more

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Dominoes and Mathematical Induction

September 27, 2010 GB College Mathematics, Number Theory

Dominoes are Falling Down

If you queued ten thousand dominoes on a very long table and you want to let them all fall just by letting the first domino fall, then how would you queue it?

The best idea probably is to queue them such that:

When the first domino falls, it will hit the second domino.
Make sure that each domino will hit the domino next to it and that each hit domino will fall.
If conditions (1) and (2) are satisfied, then all the dominoes will fall.

The domino effect

In fact, no matter how many dominoes we put on the table, as long as conditions (1) and (2) are satisfied we are sure that all the dominoes will fall. » Read more

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Young Gauss and the sum of the first n positive integers

September 15, 2010 GB College Mathematics, High School Mathematics, Number Theory

Carl Friedrich Gauss was one of the most prolific mathematicians of all time. In fact, he was considered by many as the “Prince of Mathematicians” because of his numerous contributions in different fields of mathematics.

Gauss displayed his genius at an early age. According to anecdotes, when he was in primary school, he was punished by his teacher due to misbehavior. He was told to add the numbers from 1 to 100. He was able to compute its sum, which is 5050, in a matter of seconds.

Old Gauss

Now, how on earth did he do it? » Read more

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