Everule's blog

By Everule, history, 6 weeks ago, This is a blog starting from the very basics of number theory, in a way that flows fluidly from one concept to another and is based in developing an intuitive feeling for the basics of elementary number theory. This is not a blog to simply gloss over. I consider more of a guided exploration into the world of discovering things in the world of number theory, and I don't expect anyone to immediately understand all the insights in this blog. But if you put an honest effort into discovering how I find these insights you will find much use for my blog.

If you do not know some notation or some elementary theorem I use you should refer to this.

Elementary definitions

Greatest common divisor, Additive structure of residues mod n, and Bezout's Theorem

Multiplicative structure of residues mod n and Fermat's little theorem

Chinese remainder theorem and linear equations modulo n

Fundamental Theorem of arithmetic

Extended Chinese remainder theorem

Multiplicative functions and Mobius inversion

Primitive roots and modular logarithm

Probabilistic primality test Comments (5)
 » Auto comment: topic has been updated by Everule (previous revision, new revision, compare).
 » 6 weeks ago, # | ← Rev. 4 →   [Deleted]
 » Really nice blog; it does a great job considering the amount of material it covers. Hoping to see similar stuff for other topics too. Minor nitpicks In the elementary definitions part you probably meant $0 \le r < |k|$ instead. In the definition of multiplicative functions and convolution, it is usually understood that the arguments of the function are positive integers (and the same goes for the divisors over which the sum is taken). Not a nitpick per se, but it would be nice to also mention the form of the numbers that have a primitive root; personally it was a bit weird to see the number of primitive roots and the density versions of the PNT mentioned but not this fact. The second limit looks like it has a typo, I think it should be $\lim_{n \to \infty} \frac{\pi_{x, m}(n)}{\frac{1}{\phi(m)} \frac{n}{\log n}}$ instead of $\lim_{n \to \infty} \frac{\pi_{x, m}(n)}{\phi(m) \frac{n}{\log n}}$. Also it might be better to mention that $\log$ is the natural logarithm here. "Essentially the primes are equally distributed among every element in the RRS of some n." This is slightly inaccurate; the only thing that follows is that they have equal density. However, to my knowledge there are no results that talk about even the difference being bounded. The Chebyshev bias is a nice point to start. The last paragraph for probabilistic primality testing is a bit vague and perhaps ambiguous, and it might be better to give references to the Miller Rabin primality test.
•  » » Thanks for the corrections, I've updated all of those, and added some more natural proofs to make it easier to understand now. Also the Chebyshev bias is new to me, I just assumed it makes sense for all the prime count to be independent of which element of the RRS it is part of.I would write another wall of text for another topic, However I don't know anything I could write this much about, most other concepts are quite bounded in how much you can explain about them. If you or anybody else have an idea I'll be happy to explain it in this form.
•  » » » Just one more correction: the claim about $p^k$ having a primitive root doesn't hold for $p = 2, k = 3$ (it holds for odd $p$, and the only powers of 2 having a primitive root are $2, 4$).