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This will be a post illustrating how unique factorization in the ring helps us solve the diophantine equation . We first start by proving Lemma: The ring is a UFD. Proof. Define . The identity proves that So is a … Continue reading
What is an example of a compact space that is not sequentially compact? Let’s recall that: A space is compact if every open cover of has a finite subcover. And is sequentially compact if every sequence in has a convergent … Continue reading
Recently I have been browsing the book “Topics in Group Theory” by Geoff Smith and Olga Tabachnikova. This is a great book to learn advanced group theory at the undergraduate level. I only wish I knew this book earlier in … Continue reading
Perhaps the fastest proof of uncountability of is using the diagonalization trick. However, there’s another proof that I learnt from Topology by Munkres (Theorem 27.7 in page 176) which is nice in its own way. It is enough to show that … Continue reading
Goldbach’s Conjecture states that every even integer greater than can be written as sum of primes. This problem is notoriously difficult, and still open. However, one can prove the following version of Goldbach’s Conjecture for the ring : Proposition. Let … Continue reading
Many people have told me that CayleyHamilton Theorem is their favourite theorem in linear algebra. I also like this theorem a lot (Proof: I am dedicating a blog post about it!) Let be a linear transformation on a finitedimensional vector … Continue reading
I think the following is a good exercise for first course in real analysis. Does there exist continuous bijection from to ? (1) Well, it feels like the answer should be “no”. At … Continue reading