Number Theory

https://www.youtube.com/watch?v=SLyVAR3JV5A Imagine that pieces of chicken are sold in groups of 3 and groups of 7. What number of pieces of chicken are achievable as non-negative integer combinations of 3 and 7, and how many numbers are impossible to achieve? This is the Frobenius coin problem. When there are two coprime denominations, the answer is […]

https://www.youtube.com/watch?v=txEekc2Rd8U Similar to the divisibility rule for division by 9, there is a trick for division by 11 in base-10: we take the sum of the digits with alternating signs, and this quantity is divisible by 11 if and only if the original integer is divisible by 11. In this video, we prove that this

https://www.youtube.com/watch?v=AuBLVE9n1ew Pierre de Fermat said that his most prized technique in number theory was infinite descent. We describe infinite descent as an application of the contradiction method to the well-ordering principle. Then we give an example of applying infinite descent to a Diophantine equation.

https://www.youtube.com/watch?v=oIQe9SLi39U Sooner or later, almost every student comes across the most famous of mathematical theorems, the Pythagorean theorem. This theorem is encapsulated by the equation which states that a square is split additively into two squares. One may then ask for all triples of positive integers that satisfy this equation. We can uniquely parametrize all

https://www.youtube.com/watch?v=eu9ysTIsmoU Around 300 BC, the following problem was posed in the ancient Chinese book Sun-tzu Suan-ching: “There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?” We

https://www.youtube.com/watch?v=7uJBhr-Hwxc Inspired by the identity in Bézout’s lemma, we might ask for all integer pairs $(x, y)$ that satisfy the equation $ax+by=c$ for a given integer triple $(a, b, c)$. Firstly, we address the question of when a solution exists at all, and secondly we show how to generate all solutions from one solution. This

https://www.youtube.com/watch?v=PybffOyC66I One of the most accessible methods for solving Diophantine equations (for us, these are multivariable equations where we want only integer solutions) is fudging and factoring. In essence, we rewrite the equation so that there is a factored expression with variables on one side and the other side is a constant. Then we can

https://www.youtube.com/watch?v=7ywANWEsxLw Fermat’s Little Theorem, also known as FLT but not to be confused with Fermat’s Last Theorem, is a special case of Euler’s congruence, which we have addressed in a different video. Instead of deriving Fermat directly from Euler, we go through the proof of Euler specifically in the case of a prime modulus in

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