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### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},$$ where the ...
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### If $2^x$and $3^x$ are integers, must $x$ be as well?

I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number. If $n^x$ is an integer for ...
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### Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...
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### An inequality for cosine of n

Can anyone provide a proof of the following inequality? If $n$ is a positive integer, $n\geq2$, then $$\cos(n) \leq 1 - 2^{-n}.$$ This is satisfied if $n$ is not within about $2^{-n/2}$ of a multiple ...
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### How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...
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### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
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### When is $n/\ln(n)$ close to an integer?

As usual I expect to be critisised for "duplicating" this question. But I do not! As Gjergji immediately notified, that question was from numerology. The one I ask you here (after putting it in my ...
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### Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result: If $\alpha$ is a real ...
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### Diophantine approximation in the Julia set

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...
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### Should the number of small solutions in Roth's theorem be bounded uniformly, assuming the target is an algebraic integer?

Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible ...
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### Distribution of polynomials mod 1 using co-prime integers

If $a$ is any real irrational, then the set of numbers of the form $ax+y$ with $x$ and $y$ co-prime integers is dense in $\mathbb{R}$. I managed to prove this, in what I suspect is an overly ...
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### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...
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### Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
Given any integer $n$ sufficiently large, I want to prove (or disprove) that there exists another integer $m\ge n$ with the form $m=2^a3^b$ ($a,b$ are no negative integers) such that $m-n=o(n)$, i.e., ...