Plan:
- Questions
- Wiki participation
- Recall (modern) characterization of Pythagorean triples
- Read Euler's proof that $x^4 + y^4 = z^4$ is impossible
Student Questions:
- Homework (proof of perfect number with grouping up factors of 496 as a hint. I made some conclusions but am still kind of confused. I would like to solve the problem though because I have done some work on it) - Rob
The discussion page can be found here, for observations and remarks. Edit this page to provide a single coherent narrative of what went on in class.
I wrote an awk program to count the number of times each of you edited a wiki page (consecutive changes to the same page don't count). Here's the result.
| Nicole Nocera | 7 |
| Mingzhi Liu | 9 |
| Mou | 1 |
| Shira Kaminsky | 6 |
| yawouz | 2 |
| PowellVacha | 2 |
| Lin Han | 13 |
| Angela Rogers | 5 |
| robmoray | 21 |
| jun chen | 3 |
| Rachel Bay Chaparro | 1 |
| Matt Lehman | 6 |
Remember that your grade will depend (in part) on your contributions to this collective endeavor.
• Hint for proving $2^{n-1}(2^n-1)$ is perfect number when $2^n-1$ is prime.
• Why is 28 a perfect number?
$28=4*7$.
Now separate the six divisors of 28 this way and sum separately:
• Why is 496 a perfect number? Try the trick above. If you use a trick twice it becomes a technique.
Converse: if you have an even perfect number, it looks like $2^{n-1}(2^n-1)$. This is not as straightforward.
Some of Euler (“oil er”)’s contributions
• $e^{i\pi}+1=0$
• $\frac{\pi^2}{6}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}…$
• $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}…$ diverges
It takes a little work to make Euler’s proofs rigorous by modern standards. During Euler’s time, there was no concept of ($\epsilon$ and $\delta$)
• Quadrate = square
• Biquadrate = fourth power
• Word “quad” means square
Went through the book of how Euler proved that $x^4+y^4=z^2$ has no nontrivial solutions.
"On some formulas of the form $ax^4+by^4$ which are not reducible to a square" The proof would refer to Euclids proof to explain in more details.