1
vote
2answers
54 views

Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
0
votes
5answers
58 views

How to illustrate that

there is $a, n \in \mathbb Z^{+}$ and prime number $p$, with relationship: $$p|a^{n}$$ It's straight forward that $p|a$, but I can't find a proper illustration of it.
-2
votes
3answers
135 views

An intutive way to think about odd and even numbers. [closed]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
6
votes
1answer
164 views

What is the big picture behind AKS algorithm?

Despite a number of question on AKS algorithm here, there does not seems to anything related to the idea behind it (for those who don't know, AKS primality testing is found in PRIMES is in P). I read ...
8
votes
1answer
129 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
18
votes
1answer
320 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
4
votes
1answer
136 views

Intuition behind the Jacobi triple product

Jacobi's triple product identity states that: $\displaystyle \sum_{n = -\infty}^{\infty}z^{n}q^{n^{2}} = \prod_{n = 1}^{\infty}(1 - q^{2n})(1 + zq^{2n - 1})(1 + z^{-1}q^{2n - 1})$ I've seen a messy ...
36
votes
5answers
1k views

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
2
votes
2answers
212 views

How do you find the closed form of these recurrence relations?

I found these two recurrence relations in an old textbook and was hoping someone could show me how to solve them for their closed form. If not, a final answer would also be appreciated, as it helps ...
8
votes
5answers
376 views

Arithmetic of irrationals and the Vedanta behind it..

I am really curious about the Vedanta behind the arithmetic operations on irrational numbers. It still got aggrevated after the productive discussions with my friend. So I decided to ask it here. ...
10
votes
1answer
330 views

What is the intuitive meaning of “conspiracy” in number theory?

Assuming very little number-theoretic background from my part, could you please explain me what is the intuitive meaning behind "conspiracy" in number theory? There is no formal entry on Wikipedia and ...
4
votes
1answer
352 views

Subset of natural numbers such that any natural number except 1 can be expressed as sum of two elements

Let $X$ be the set of natural numbers $k_i$, $k_i \geq 1$, with the property that at least one of the equations $p_i = $6$ k_i \pm 1$ gives the $i$-th prime number (disregarding $2$ and $3$), and ...
13
votes
1answer
1k views

What is the intuition behind the concept of Tate twists?

For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
33
votes
1answer
685 views

How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
4
votes
1answer
452 views

Largest known integer

Does there exist a property which is known to be satisfied by only one integer, but such that this property does not provide a means by which to compute this number? I am asking because this number ...
15
votes
3answers
804 views

What is the intuition behind Gauss sums?

Let $ \chi $ be a character on the field $ F_p $, and fix some $a \in F_p $. We define a Gauss sum to be: $g_a (\chi) = \sum_{t\in F_p}\chi(t)\zeta^{at}$ where $\zeta$ is a primitive $p^{th}$ root of ...
21
votes
4answers
988 views

You are standing at the origin of an “infinite forest” holding an “infinite bb-gun”

I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes... You are standing at the origin of an ...