Prove that negative Pell’s equation has no solutions if the period length of continued fraction of $sqrt{D}$ is even and infinite solutions otherwise

Prove that negative Pell’s equation $x^2-sqrt{D}y^2=-1$has no solutions if the period length of continued fraction of $sqrt{D}$ is even and has infinitely many solutions if the length is odd. I feel if the length is odd then the solution might have something to do with the convergent of $sqrt{D}$. Any further hints?

Finding z-transform for the probability of winning and partial fraction expansion

Can someone please provide feedback on my solutions and help with the last part? Thank you for your time and help!

Two people, A and B are playing a game and A has a 60% chance of winning, A gains +1 point on a win. There is a 15% chance of a tie; A loses one point on a tie. A has a 25% chance of losing; A loses two points on a loss.

If A reaches +3 points, A wins match. If A reaches -3 points, B wins match.

a. What is the state transition matrix? Going from k games to k+1 games?

$textbf{Solution:}$ I just took the first markov chain, and we get $begin{pmatrix} 0.46 & 0.54\ 0.34 & 0.66end{pmatrix}$.

b. What is the chance that $A$ will win the match assuming they start out at even. A has zero points?

$textbf{Solution:}$ The $p(A|text{win}) = 0.6^3 = 0.216$

c. What is the z-transform for the probability of $A$ winning after k games?

$textbf{Solution:}$ $=frac{1}{1-0.6z^{-1}}$

d. From the z-transforms, determine the probability of $A$ winning the match, i.e., the partial fraction expansion over the (z-1) term?

number theory – Continued Fraction Factorization Algorithm please explain with Code

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contest math – I don’t understand how to reduce this fraction to the stated solution:

The fraction is as follows:

9 x 11 + 18 x 22 + 27 x 33 + 36 x 44 /
22 x 27 + 44 x 54 + 66 x 81 + 88 x 108

That’s all fine. What I don’t get is that my textbook says this reduces to the following:

9 x 11 + (1² + 2² + 3² + 4²) / 22 x 27 x (1² + 2² + 3² +4²)

I don’t understand how the sum of consecutive squares can be deduced from that fraction, or why the denominator contains “22 x 27 x “ as opposed to the numerator which is “9 x 11 +”

Any insight would be really appreciated!

Google Sheets value on chart off from source by very small fraction

Trying to figure out why Google Sheets chart is off from the source data by a small fraction. My sheet has 7 rows of data and a sum row. All data only has 2 decimal points and the sum obviously has 2 decimal points. Yet when plotted, the sum is off as shown in the screenshot below.


The callout on the chart is pulling its data from cell C12. Formula for C12 is just SUM(C4:C10). I expanded the decimal places in C12 to show that nothing was hidden/rounded/truncated. Trying to understand why this is happening and how to stop it.

What is the probability of mining a block by a miner controlling a fraction of hashing power?

Suppose a miner controls α fraction of the hashing power and the chain on which he is trying to mine has the mining rate given by f. How to find the probability of mining a block by the miner?

How do i type a fraction on my windows 10?

I am not sure how to type a fraction on my windows ten because i need to do homework and i can not work out how to type a fraction.

How can we generate this more general form of the continued fraction?

How can I write Mathematica code for this continued fraction with alternating terms?

How can we generate this more general form of the above-continued fraction?


mutational algebra – Calculate the fraction field of formal series over integers

What is the fraction field? $ K $ domain $ mathbb Z ((X)) $?
It is strictly smaller than the Laurent series field. $ L = operatorname {Frac} mathbb Q ((X)) $, as $ sum_ {i geq 0} frac {X ^ i} {i!} in L setminus K $.
In fact, a series of Laurent $ f (X) = sum_ {i in mathbb Z} q_iX ^ i in L $ must have all its coefficients $ q_i in mathbb Z ( frac 1n) $ to belong to $ K $ ( where $ n $ is an integer that depends only on $ f (X) $)
But is that necessary condition sufficient?

Integration: integral of an expression that includes a fraction that has modified the Bessel functions of the first type in both the numerator and the denominator.

I am looking for an analytical result of the following integral
$$ iint_0 ^ infty {{ rm {d}} x { rm {d}} y {x ^ 2} {y ^ 2} exp left {{- { alpha _1} {x ^ 2} – { alpha _2} {y ^ 2}} right } frac {{{ rm {I}} _ 1 ^ 2 left ({ beta xy} right)}} {{{{ rm {I}} _ 0} left ({ beta xy} right)}}} $$
where $ { rm {I}} _ 0 ( cdot), { rm {I}} _ 1 ( cdot) $ are the modified zero and first order Bessel functions of the first type, respectively, and $ alpha_1, alpha_2, beta $ are positive constants

I have referred to a book called "TABLES OF SOME INTEGRAL INTEGRAL FUNCTIONS OF BESSEL'S INTEGRAL ORDER", but I did not find a solution.

It will be highly appreciated if there are any ideas.