February 28

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:

(1)
\begin{equation} (1 + 2 + 4) + 7(1 + 2 + 4) \end{equation}

• 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.

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