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https://www.youtube.com/watch?v=07x2r1-1nmA Given a modulus and an integer coprime to it, we want to find a power of that integer that is congruent to 1 in that modulus. One reason is that this allows us to find an explicit multiplicative inverse of that integer. This is where Euler’s congruence comes into play. Euler gives us, not […]

https://www.youtube.com/watch?v=6RXbyueyov8 Diophantine equations sometimes have no solutions. In such cases, it might be possible to reduce the polynomial equation to a certain modulus and show that no combination of residues for the the variables produces the residue on the other side. There is no official name for this method, but we call it the “modular

https://www.youtube.com/watch?v=BghI44RQe-A What do we get if we fix a positive integer, and take the sum of Euler’s totient function applied to each positive divisor of the original integer? Incredibly, it returns the original integer. This means that the summation function of Euler’s totient function is the identitiy function. We prove this fact using an out-of-the-blue

https://www.youtube.com/watch?v=C8i209klzmA The Möbius inversion formula allows us to recover a function from its summation function. A fair amount of machinery needs to be built (or in our video, assumed) to prove the inversion formula, but the result is well worth it. For example, it can be applied to derive an explicit formula for the Euler

https://www.youtube.com/watch?v=BPCIBhoLeOo Given a positive integer, we might wonder about the sum of its positive divisors or factors. This is called a divisor function. The answer lies within the original integer’s prime factorization. We call this function the sigma function. Moreover, there is a second form of this function which we can derive using the formula

https://www.youtube.com/watch?v=LdvtBEM5iEw Wilsons theorem, in its most complete form, asks for the remainder when $(n-1)!$ is reduced modulo $n$. Surprisingly, we can completely classify the remainders according to three cases of $n$: prime, the number $4$, and otherwise. We state and prove this classification in this video.

https://www.youtube.com/watch?v=UzQkHMgaEqU Often times, our teachers ask us to “reduce the fraction” to “lowest terms.” But why does a least representation exist and in what sense is it unique? In this video, we prove the existence and uniqueness of reduced forms of fractions. Moreover, we show that the least representation generates all other representations.

https://www.youtube.com/watch?v=OOpXrQsKLqg In math contests and olympiads, there is a variety of lemmas surrounding the idea of dividing a power. These can come in handy quite frequently. We decided to make a video that compiles the most common and useful of these lemmas without proof. Enjoy! May these lemmas serve you well.

https://www.youtube.com/watch?v=yQiMrmHxwhA Most of us have used or at least heard of prime factorizations. But why do they exist and why are they unique? This is called the fundamental theorem of arithmetic, and we prove it in this video. Our main technique will be strong induction, along with casework on whether the number is a prime

https://www.youtube.com/watch?v=3jXkvF7bPXs Given a positive integer, we might wonder how many positive divisors or factors it has. Interestingly, we can count the number of positive factors using the prime factorization of the original integer. This counting function is called the tau function, denoted by the Greek letter tau. In this video, we derive the formula for