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Let $\alpha$, $N$ be given. By Dirichlet's approximation theorem, there exist infinitely many integers $p,q$ such that $$ |q\alpha - p| < \frac{1}{|q|}. $$ Therefore, there exist infinitely many integers $p,q$ with $|q| > N$ such that $$ |q\alpha - p| < \frac{1}{|q|} < \frac{1}{N}. $$


Let $[x]$ denote the largest integer not exceeding $x.$ Let $e \in R-Q.$ For $k\in N,$ define $d_k= e k-[e k]$ and define $k'$ as follows : Let $l_k\in N$ where $l_k d_k<1<(1+l_k)d_k.$ Let $k'=-l_k k$ if $1-l_k d_k<(1+l_k)d_k-1.$ Otherwise let $k'=(1+l_k)k.$ $$\text {Observe that }\quad 0< d_{k'}<d_k/2.$$ Let $k_1=1$ and let ...


This is not a complete answer, but may be useful. The largest root of the polynomial $$269x^2-503x+209$$ is $$\frac{17+15\sqrt{5}}{7+15\sqrt{5}}$$ Changing the polynomial to $$(25)^2\times269x^2-25\times63\times 503 x+63^2\times 209$$ modifies the root to the approximation given. $$\pi\approx\frac{63}{25}\times\frac{17+15\sqrt{5}}{7+15\sqrt{5}}$$ This ...


Continuing from previous answers and comments, solving for $x$ equation $f(x)=x^k-a$ (for any power $k$), can be achieved with your criteria using Newton method or, faster, with its variants such as Halley, Householder or even higher order methods they do not have a name). For example, using Halley method, $$x_{n+1} = x_n - \frac {2 f(x_n) f'(x_n)} {2 ...

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