## 9. dummyoutersection

### 9.4. Sample Article

This "article" is a demonstration of how one can write mathematics in Noteshare. Use arbitrary LaTeX inside the traditional dollar sign, bracket delimiters, and a form of the LaTeX environment. Equation numbering and cross-referencing is supported, as is the numbered theorem environment.[1]

Be sure to choose **Source** from the **View** menu
to see how this article was written.

Observe the footnote reference in the previous paragraph. The footnote is *way*at the bottom. However, you can click on it it reference it.

#### 9.4.1. Pythagorean triples

One of the first real pieces of mathematics we learn is this:

*Let \(a\), \(b\), and \(c\) be the sides of a right triangle, where \(c\) is the hypotenuse. Then \(a^2 + b^2 = c^2\).*

The Pythagorean theorem suggests an equation,

\[ x^2 + y^2 = z^2 \] | (1) |

If we demand that the unknowns be integers,
then this is a *Diophantine equation*. We all
know one solution, to equation (1) the
3-4-5 triangle. However, there are many
more, in fact, infinitely many. One
way of generating more solutions is to rescale existing ones.
Thus 6-8-10 is a solution. However,
what is interesting is that there are infinitely
many *dissimilar solutions*. One is 5-12-13.
Quite remarkably, there is a clay tablet (Plympton 322) from
Mesopotamia, dated to about 1800 BC, that
contains a list of 15 such
"Pythagorean triples." See Wikipedia.

It is unlikely that the Babylonian mathematicians, despite their sophistication and skill, knew that equation (1) has infinitely many solutions. This fact is equivalent to the statement that the unit circle centered at the origin has infinitely many points with rational coordinates.

For more information on Pythagorean triplets, see Wikipedia.

#### 9.4.2. Another result from ancient times

*There are infinitely many primes.*

Suppose that there are only finitely many primes, say \(p_1, p_2, \ldots, p_N\). Let

\[ Q = p_1p_2 \cdots p_N + 1 \] | (2) |

This number is is greater than the greatest prime, \(p_N\).
Therefore it is composite, and therefore it is divisible
by \(p_i\) for some \(i\). But the remainder of \(Q\) upon
division by \(p_i\) is 1, a contradiction. **Q.E.D.**

#### 9.4.3. The work of Fermat

Let us consider Theorem 1 once again. Pierre Fermat asked whether this family of Diophantine equations \(x^d + y^d = z^d\), where parameter \(d\) is greater than two, have any other than the obvious solutions, e.g., \(x, y, z = 1, 0 ,1\) where one variable is zero. Fermat conjectured that the answer was no, and he wrote in the margin of Diophantus' treatise that he had a marvelous proof of this fact, alas, too long to fit int he space available. More than 300 years later, Andrew Wiles, using techniques developed only in the twentieth century, gave a proof of the theorem of Fermat.

#### 9.4.4. The work of Georg Cantor

Georg Cantor introduced the notion of *set*, e.g.,

\[ \NN = \set{ \text{integers} } = \set{ 1, 2, 3, \ldots } \] | (3) |

Story to be continued …

However, let’s check that our cross-references work. We learned about the Pythagorean formula (1), and the number of solutions it has in Theorem Theorem 1. We also learned in Theorem Theorem 2 that prime numbers are quite abundant in Nature: there are infinitely many of them. Our last, incomplete section, featuring the set of integers (3), is devoted to the work of Georg Cantor.

Created October 2, 2014, last updated: August 21, 2015