# Tagged Questions

A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

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### Convergents of continued fractions

Let $d$ and $m$ be positive integers such that $d$ is not a square and such that $m\leq\sqrt{d}$. I want to prove that if $x$ and $y$ are positive integers stafisfying $x^2-dy^2=m$ then $x/y$ is a ...
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### How to calculate remainder value of a fraction

Question: Four brothers split a sum of money between them. The first brother received 50% of the total, the second received 25% of the total, the third received 20% of the total, and the fourth ...
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### Find the real number $x$ represented by continued fraction $[12;2,2,12,2,2,12,2,2,12\dots]$

I need to fins the real vlaue of x for the continued fraction (Image attached) I have tried partial coefficient method, but didn't get the exact answer. I there any way where we can identify the ...
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### Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So ...
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### Two similar integration about continued fractions

Prove that \begin{align*} \int_0^{+\infty} \cfrac{\sin nx}{x + \cfrac{1}{x + \cfrac{2}{x + \cfrac{3}{x + \cdots}}}} \, dx &= \cfrac{\sqrt {\cfrac{\pi }{2}} }{n + \cfrac{1}{n + \cfrac{2}{n + \cfrac{...
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### Are the numerator and the denominator of a convergent of a continued fraction always coprime?

Is it true that for a convergent $A_k = p_k/q_k$ of the continued fraction expansion $[a_0;a_1,a_2,\dots]$ the numerator $p_k$ and the denominator $q_k$ are always coprime? If yes, how would one show ...
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### Evaluation of a continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
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### a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
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### Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
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### Why is this continued fraction expansion what it is?

We have to find the continued fraction expansion of the roots of $1553 t^2 + 6014 t + 5820 = 0$, that is, $(\sqrt{14356} - 6014) / 3106$ Simplifying, $(\sqrt{3589}- 3007) / 1553$ The continued ...
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### a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
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### Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ \$x^2+\frac{1}{x^2}...