# Tagged Questions

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### 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 ...
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### 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.
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### 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...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...