February 26 Discussion

Here is a link to Euclid's proof of the non-finitude of the prime numbers. While reading, keep in mind that Euclid's use of the word "measured" corresponds to our "divided". His method differs from the way I remember learning the proof, where a number is constructed as the product of the first n primes + 1, the nth prime being the greatest. Then either that number is prime, or is a product of some prime greater than the greatest prime. So by contradiction, there is no greatest prime. Instead Euclid essentially says, given a certain number of primes, one can always find another, which is a simpler argument.

His approach demonstrates the rejection of the concept of an actual infinity, which took 2300 years to be legitimized by Gödel in the late 1800s, which I hope we'll get to cover later in the course.


Note that I did add a more modern proof of the infinitude of primes theorem. For those looking for this proof, it can be found here. For those looking for a more comprehensive collection of Euclid's elements, Clark University (of Worcester, MA) hosts on one of its virtual servers what appears to be a complete collection of Euclid's Elements available to read online. Feel free to check this out here, and note that you can navigate through Elements using the drop-down box at the bottom of the landing page. Props to those that know the mathematical significance behind the subdomain of www.clarku.edu that this is listed under.

I learned the proof for the infinitude of primes in a way that more closely resembles Matt's (above). However, as Euclid's proofs have begun to make more sense as we have progressed to number theory, with the algebraic aspect providing more comfort in working with the geometrical representations that the Greek's were accustomed to, I do actually quite like Euclidean proof that we covered in class.

I did have a chance to take a look at this theorem, however, which does have a Wikipedia entry. While the topological proof concept does look highly interesting, I have absolutely no experience with topology, and probably would not do so well with understanding the proof. Euler's proof (he is another respected historical mathematician, at least from my perspective) would be interesting to examine if we have time, as his use of a series coupled with the Fundamental Theorem of Arithmetic (see the article) was quite interesting.

EDIT: One more interesting fact: According to Wikipedia, the largest perfect number was discovered this year (2013) by Cooper, Woltman, Kurowski, et al. It is 34,850,340 digits long and is the 48th perfect number discovered thus far. Woltman and Kurowski helped to discover perfect numbers 37-48, though only Woltman was on the team that discovered the 36th perfect number. Also, the Cooper that discovered this year's perfect number is also the Cooper who found the most recently discovered Mersenne Prime via GIMPS.

— Rob

  • Of course it's the same Cooper, since it's the same discovery - Euler proved the one to one correspondence between Mersenne primes and even perfect numbers. Ethan

Should I put my class note on the discussion page or on the lecture page?? My notes has been "reformatted" twice for the last two classes; some of my stuff were left, but my name wasn't acknowledged.

  • I think it's fine when people rework other people's contributions on the lecture notes, since that page is supposed to be a record of what was said in class. Wikidot records page changes, so I know you have contributed. Use this page for (signed) comments with your observations and remarks. Ethan.

Another way to look at Euclid's Proposition 29, Book X is to consider the plane numbers ab and cd where $\frac { a }{ b } =\frac { c }{ d }$, and neither ab nor cd are odd. The Pythagorean Triple resulting from these two plane numbers are given as:

\begin{align} { \left( ad \right) }^{ 2 }+{ \left( \frac { \left| ab-cd \right| }{ 2 } \right) }^{ 2 }={ \left( \frac { ab+cd }{ 2 } \right) }^{ 2 } \end{align}

Since $ad=bc$, the value of $ad$ in the first parenthesis can be replaced with $bc$.

For example, take the plane numbers 18 and 32. These are plane numbers because their factors are in the proportion $\frac { 3 }{ 6 } =\frac { 4 }{ 8 }$. If we plug them into the equation above, we get:

\begin{align} { \left( 3\cdot 8 \right) }^{ 2 }+{ \left( \frac { \left| 18-32 \right| }{ 2 } \right) }^{ 2 }={ \left( \frac { 18+32 }{ 2 } \right) }^{ 2 } \end{align}
\begin{align} { \left( 24 \right) }^{ 2 }+{ \left( \frac { \left| -14 \right| }{ 2 } \right) }^{ 2 }={ \left( \frac { 50 }{ 2 } \right) }^{ 2 } \end{align}
\begin{align} { \left( 24 \right) }^{ 2 }+{ \left( 7 \right) }^{ 2 }={ \left( 25 \right) }^{ 2 } \end{align}

It is also important to note that this is a primitive Pythagorean triple. The 3-4-5 triple is obtained by using the plane numbers 2 and 4. (Powell)

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