We prove Fermat’s Christmas Theorem using Gaussian integers, linking sums of two squares to the solvability of \(x^2\equiv_p -1\).
We explain why, if Fermat had a proof, it should have been very remarkable.
We introduce modular arithmetic and understand the meaning of Chinese Remainder Theorem.
We will prove how some interesting propositions follow from the Chebotarev density theorem.
Expository tour of Euclid’s proof that there are infinitely many primes, preceded by a discussion of Euclidean division and the fundamental theorem of arithmetic.
We investigate when injectivity is sufficient for surjectivity in an algebraic setting. We first treat the case of an algebraically closed field and give a complete proof, then discuss generalizations to schemes.
We introduce Fermat’s Christmas Theorem and try to understand its meaning through concrete examples.