# Tagged Questions

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### Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
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### Division algorithm for the natural numbers.

I am trying to prove the following statement from Tao's analysis book. Definition of multiplication $ab++=ab+b$. Definition of addition $(a++)+b=(a+b)++$. Let $n$ be a natural number, and let $q$ ...
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### Curious set of $n\sin^2(n)$

I consider this set $S_{0}=\{ n \in \mathbb{N}: n\sin^2(n) < 1 \}$. And I have some questions. See the elements of $S_{0}$= $\{ 1,3,6,19,22,25,44,47,66,69,88,110,132,154,157,176,179,$ (common ...
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### Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...
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### Questions for first year students at the University. [closed]

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is ...
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### How would one find a) All the primitive characters modulo 8, b) All the non-primitive characters modulo 8?

Preferably explained in novice terms! I can start it off by having the multiplicative group modulo 8 with elements $[1], [3], [5], [7]$ and not sure where to go now. I see there is a similar question ...
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### How prove the constant term of $\left(1+x+\frac{1}{x}\right)^p\equiv1\pmod {p^2}$

if $p>3$ is odd prime number,show that: the constant term of $$\left(1+x+\dfrac{1}{x}\right)^p\equiv1\pmod {p^2}$$ My try: since ...
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### $\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
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### Number of lattice points in an annulus

Consider the lattice spanned by two nonzero complex numbers $\xi_{1}$ and $\xi_{2}$ such that their ratio is not real. Let $w = m\xi_{1} + n\xi_{2}$. Let $A(n)$ be the number of lattice points such ...
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### What is the convex-hull of the set $\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2$

I know that set $$E=\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2$$ has infinitely many points on the line $y=x-1$, which suggests this line to be included in the upper part of the ...
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### Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
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### Berkovich analytifications and non-archimedean geometry of Transseries

In Transseries and Real Differential Algebra by Joris van der Hoeven it is said that Transseries admit a rich non-Archimedean geometry (somewhere on page 13), but since the book isn't about that, ...
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### Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
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### Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
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### How find this$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+…+\frac{1}{{{p}_{n}}}<10$

Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$$ This problem is from this ...
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### Absolute convergence of Euler products and infinite series

We know that given a multiplicative function $f$ for which the series $\sum_{n=1}^\infty f(n)$ converges absolutely then so does the Euler product $\prod_{p}\sum_{k=0}^\infty f(p^k)$, but does the ...
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### Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
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### Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
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### Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
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### A product of two sums of four squares

I am dealing with a problem and I hope you can help me. I have already proved this: Let us suppose that integers $m$ and $n$ can be written as sum of squares of two integers. Prove that m*n can also ...
I admit that I'm probably out of my depth with this question, but I can't help but feel curious. I wanted to show that, in the sequence $\{\sin(n)\}$, there is never a largest term (the sequence ...