https://www.youtube.com/watch?v=4p68aPqeegA We can ask the question: how many times does a prime $p$ divide into a factorial $n!$ until $p$ does not divide $n!$ any more? This is called the $p$-adic valuation or order of $p$ in $n!$. Interestingly, there is a nice formula for this called Legendre’s formula, which relies on a counting argument. […]
Legendre’s Factorial Formula Read More »
https://www.youtube.com/watch?v=mOvVImV0GSo A set is called countable if there exists a bijection from the positive integers to that set. On the other hand, an infinite set that is not countable is called uncountable. In this video, we show that the real numbers are uncountable using a method par excellence called Cantor diagonalization. There are also some
Reals are Uncountable by Cantor Diagonalization Read More »
https://www.youtube.com/watch?v=pyctG41q9os A set is said to be countable if it can be put in bijection with the positive integers. It is a remarkable fact that a set with more than “double” the number of elements, namely the integers, is countable. Even more remarkable is that the rational numbers are countable. We prove the latter in
Rationals are Countable by Cantor Snake Read More »
https://www.youtube.com/watch?v=_h3nyoP4TKg For those familiar with Euclid’s proof of the infinitude of prime numbers, this video will be a treat. All odd primes leave a residue of $1$ or $3$ modulo $4$. With a slight modification of Euclid’s proof, we can show that there exist infinitely many primes congruent to $3$ modulo $4$, which is what
Infinitely Many Primes 4n+3 Read More »
https://www.youtube.com/watch?v=hw-A1JDuunA Triangular numbers are those numbers that are the sum of an initial segment of the positive integers. Mersenne numbers are numbers that are $1$ less than an integer power of $2$. As number theorists will, it may be asked how many Mersenne numbers are triangular as well, and what exactly is the list of
Triangular Mersenne Numbers Read More »
https://www.youtube.com/watch?v=ZjVkb63QORo Given a prime, Fermat’s little theorem gives us a certain collection of congruences modulo the prime. The converse, if it were true, would say that the truth of those congruences implies the primality of the modulus. The problem is that this simply is not true. The composite moduli that are counterexamples are called Carmichael
561 is a Carmichael Number Read More »
https://www.youtube.com/watch?v=ULm1St3FxcA Given a prime p, we show that every binomial coefficient in the pth row of Pascal’s triangle is divisible by p, unless we are talking about the entries of 1 on the far left and far right. We deduce this from a more general result, and deduce from it a property of the $p^{text{th}}$
Frobenius Endomorphism Read More »
https://www.youtube.com/watch?v=6RGUeGJScBM Normally, we are told that there are no convenient divisibility rules for division by $7$. This is actually not true. While there are no rules that are as elegant as the rules for for any other positive integer less than or equal to $12$, there are a couple of tricks that we can use
Divisibility Tricks for 7 Read More »
https://www.youtube.com/watch?v=ESHPuVdMBgw Eisenstein’s criterion gives us a sufficient criterion for a property of polynomials with integer coefficients that is difficult to capture in general: irreducibility, meaning the impossibility of factoring it with rational coefficients. We apply this criterion to a prime $p$’s cyclotomic polynomial by applying a shift, which introduces binomial coefficients from the $p^{text{th}}$ row
Eisenstein’s Criterion Read More »
https://www.youtube.com/watch?v=8vubNXfqPxA Wolstenholme’s theorem tell us that a certain congruence holds modulo the third power of a prime greater than or equal to $5$. This is difficult to establish, but we can more easily prove a weaker result: the same congruence holds modulo any prime. Our technique will be to use the binomial expansion theorem in
Weak Wolstenholme’s Theorem Read More »