Unlocking the Euler Totient Function: A Key to Number Theory & Cryptography

Discover the fascinating world of the Euler Totient Function in this video! Whether you're a math enthusiast or a student, this video breaks down the essentials of the Euler Totient Function (denoted as ϕ(n)) and its significance in number theory and cryptography.
In this video, you'll learn:
What the Euler Totient Function is and how it works
How to calculate ϕ(n) with practical examples
Key properties and applications in cryptography, including RSA encryption
Euler's Theorem and its importance in number theory
Join us as we explore the mathematical magic behind the Euler Totient Function and its real-world applications. Don't miss out on this opportunity to enhance your understanding of one of math's most intriguing concepts!
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[@GameJam230]

I've actually been exploring an interesting extension of Dirichlet's Theorem of Arithmetic Progressions and the totient funcrion over the last year or so, and I don't believe there is any information on it except one or two related problems.
See, I wanted to see if one of the statements of Dirichlet's Theorem (namely, that primes are equally dense on each progression Ax+B for a fixed A, and gcd(A,B) = 1) could extend to SEMIPRIMES on a specific form of QUADRATIC progression. I did this by creating a quadratic that could be factored into two separate linear progressions, (Ax+B-C), and (Ax+B+C).
The introduction of the third variable C was not necessary, but it gave me an interesting property of being able to represent these pairs of primes on the linear progressions with the difference between them, meaning if the twin primes problem/Polignac’s conjecture are shown true, then this theorem would further show that the pairs of primes are ALSO evenly dense on pairs of these arithmetic progressions.
I managed to get almost all the way through recreating Dirichlet's original proof, but hit a road block on the last possible step, but doing so involved creating modified totient and Dirichlet L functions to account for this introduction of C and looking at pairs of numbers instead of singular, and I can safely say after months of experimenting, they have all the same properties too.
Phi_2(B,C), which I call the semitotient function, can be defined with the same multiplicative definition as the original, such that it is the product of Phi_2(P,C), for each prime factor of B, and the definition of the function on primes depends on the relationship between P and C. If gcd(P,C) = 1, then it's P-2, otherwise P-1, and this is easy to demonstrate with some simple sets written on paper. Then for repeated factors of P, you just multiply by them like in the original as well.

[@addwithad]

@@GameJam230 very interesting concept, I hadn't heard of semi-primes before. Do you have any videos or written papers about this, or is it very much a work in progress still?

[@addwithad]

@@GameJam230 also I've pinned your comment Incase anyone wants to give you any helpful ideas

[@GameJam230]

​@@addwithad Semiprimes are actually a well-known concept, they are simply numbers which are the product of exactly two primes, such as how 15 is the product of 3 and 5. They are very relevant to the subject of cryptography, as it, combined with the totient function and Euler’s theorem makes up almost the entire basis of RSA encryption.
Semiprimes are so powerful for encryption because multiplying two primes together is extremely easy, but seeing a number and determining which two primes multiply to make it is VERY hard.

[@GameJam230]

@@addwithad But, to answer your other questions, since I realize I forgot to do that with the last reply- no, I don't have any published papers of my own. I'm actually a computer science student with only Linear Algebra 1 behind me in terms of math courses, this is all self-study as a hobby. Of course, it creates the issue that it's kind of hard to find mathematicians with the time and willingness to discuss it with, because to them I'm just that kid of the week who became obsessed with weird problems like the Riemann Hypothesis and thinks he understands the universe lol.

[@surrealreal_]

interesting function

[@c.jishnu378]

I see.