Videos

Infinitely Many Primes 4n+3

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

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561 is a Carmichael Number

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

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Frobenius Endomorphism

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 $p^{text{th}}$

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Eisenstein’s Criterion

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

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