May 7

Presentations:

Note: Discussions and commentary regarding the wiki may be found here.

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Rachel and Lin - The Chinese Remainder Theorem

Powerpoint presentation at
http://www.cs.umb.edu/~eb/370/ChineseRemainder.pptx

• The 'm' in the definition of the theorem is 'p', it was kind of a typo/misinterpret. Thanks everyone for noticing it.

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Powell - An Exploration of Euler's Sum of Powers Conjecture

Here is the link to my Google Presentation: https://docs.google.com/presentation/d/1Nvr-GGpghxB0XuqcNEY8KJwMzuZRidumfgnPvzdOuQw/edit?usp=sharing

Also, here are my two homework problems:

1. Noam Elkies used the two equations below to find infinite many solutions to contradict Euler's Sum of Powers Conjecture for $n=4$. The first solutions he found were trivial solutions where one or more of the bases were equal to zero and he did this using $(m,n)=(0,1)$. Find the two equations using Elkies's equations and $(m,n)=(0,1)$. Also, find the two equations for $(m,n)=(1,0)$ and see how they differ (if at all).

(1)

\begin{align} \left( 2{ m }^{ 2 }+{ n }^{ 2 } \right) { y }^{ 2 }=-\left( { 6m }^{ 2 }-8mn+3{ n }^{ 2 } \right) { x }^{ 2 }-2\left( { 2m }^{ 2 }-{ n }^{ 2 } \right) x-2mn \end{align}

(2)

\begin{align} \pm \left( 2{ m }^{ 2 }+{ n }^{ 2 } \right) { t }^{ 2 }=4\left( { 2m }^{ 2 }-{ n }^{ 2 } \right) { x }^{ 2 }+8mnx+\left( { n }^{ 2 }-{ 2m }^{ 2 } \right) \end{align}

2. Since Euler posed his question about whether or not there are solutions to $\sum _{ i=1 }^{ n }{ { a }_{ i }^{ k } } ={ a }_{ 1 }^{ k }+{ a }_{ 2 }^{ k }+...+{ a }_{ n }^{ k }={ b }^{ k }$ for $n>4$, there have been many findings, most of them in the past 100 years. Find examples for $n=5$, $n=7$, and $n=8$. Also, explain who found it and in what year.