Here is a PDF file that contains my read-along notes for the YouTube video
series that was made by MrYouMath on the Gamma function.
MrYouMath Gamma functions notes.
Here is a PDF file that contains my read-along notes for the YouTube video
series that was made by MrYouMath on the Riemann-Euler Zeta functions
MrYouMath Riemann-Euler Zeta functions notes.
Fermat's Little Theorem states that for some integer x and some
prime integer p
xp = x (mod p).The paper presents a short induction proof.
Modulo Inverses in number theory. How to find the multiplicative
inverse of a number modulo some other number, assuming the inverse
exists. Three examples given in detail. Knowledge of modulo arithmetic
and the Euclidean Algorithm is assumed.
Another problem for Modulo Inverses in number theory using the Euclidean Algorithm.
Modulo Inverse via Euclidean Algorithm.
Typical problems that employ the GCD.
Typical problems that employ the GCD.
Linear Congruences are just linear 'algebraic' equations taken modulo some number.
LCM problems. 'LCM' mean least common multiple. It is closely related to the GCD.
LCM Problems 1.
LCM Problems 2.
Bezout's Identity/Lemma (not Bezout's Theorem) is an important theorem
in number theory. It states that if nonzero integers a and
b have GCD of d, then
there exist integers x and y such that d = ax + by.
Euclid's Lemma is an important theorem in number theory. It states
that for nonzero integers a and b and prime number p, if
p| ab then
either p| a or p| b.
A lemma is proved within the structure of a flowchart, which helps to
reveal the logic of proof. Lemma: If given n distinct positive integers,
show that either n divides one of the numbers, or else n divides the
difference of two of the numbers, or else n divides the sum of two
of the numbers.
The Well-Ordering Principle can be used to prove the simple Mathematical Induction,
and vice versa.
Wilson's Theorem states that if p is a prime number
then (p - 1)! ≡ -1 mod p.
Wilson's Theorem with proof.
Euler's Theorem with proof. Euler'sTheorem.
ChatGPT provides a Python Program.
ChatGPT wrote a Python program
for me to find the GCD of two numbers, using the Euclidean algorithm.
Included in this article is a comparison of various other Python programs
that do the same thing, but are coded differently. Very instructive.
This use of the Chinese Remainder Theorem is all about showing how to
do problems in the realm of the CRT. In these problems, we solve for x
given three modular constraints on x by three relatively prime numbers.
The problems themselves are typical, and the first two solutions are also
typical, meaning 'standard'. The last problem employs a more 'algebraic'
approach to solve it.
The Chinese Remainder Theorem is another important theorem
in number theory. The proof is given for only two relatively prime integers.
Chinese Remainder Theorem PDF file
In this survey paper we visit some fun theorems about the Fibonacci
numbers, such as the Binet Formula and the Cassini Identity, and how to
use generating functions to obtain the Binet Formula, and other surprises.
In this paper we prove that two consecutive elements of the Fibonacci sequence
are relatively prime.