1
vote
1answer
54 views

Other Interesting solutions to $a=bq+r$? [closed]

The division algorithm says $a=bq+r$, with $r$ between $0$ and $b$. Are there interesting restrictions on $r$ using number-theoretic properties that make the equation $a=bq+r$ hold, or hold with ...
2
votes
1answer
78 views

Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
13
votes
1answer
500 views

Big-Daddy-Conjectures and Hierarchy of Mathematical Conjectures

I am interested in the Hierarchy and Connections between various different open problems in Mathematics, and the most general conjectures in various fields of Mathematics. Examples of Hierachy ...
0
votes
0answers
19 views

Reference request for papers proving the existence of a special function satisfying the following conditions

Previously in this site it has been proved that there exists at least one prime between $c_n$ and $n$ where $c_n$ denotes the $n$-th composite (see the question Prove that there exists an $m$ such ...
1
vote
0answers
43 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
1
vote
2answers
146 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
26
votes
5answers
443 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
1
vote
1answer
33 views

More values of $a$ and $D $ on conditions set by me and a way to obtain more values.

Here I define -: $\alpha=a+ \sqrt D$ and $\beta=a-\sqrt D$ Then find out values of $\alpha$ and $\beta$ satisfying- $$\alpha>1 \quad and \quad -1< \beta <1 $$ and both the variables are ...
1
vote
0answers
430 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
11
votes
5answers
296 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
12
votes
2answers
1k views

High school mathematical research

I am a grade 12 student. I am interested in number theory and I am looking for topics to research on. Can you suggest some topics in number theory and in general that would make for a good research ...
6
votes
1answer
383 views

Required reading on the Collatz Conjecture

I am currently writing a paper on 3x+1 and realized that despite having enough knowledge to work on a singular facet of the problem I lack a more broad understanding of the problem. I have seen the ...
4
votes
1answer
176 views

2-dimensional $\ell$-adic representations [closed]

In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or ...
48
votes
17answers
5k views

Different ways to prove there are infinitely many primes?

This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show ...
1
vote
0answers
109 views

Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
80
votes
29answers
25k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?