I'm trying to practice finding the fundamental solution of Pell Equations, but have not found efficient ways of finding fundamental solutions. However, it seems that using  would provide an effective way of finding these solutions, according to the link. To better understand how to actually use continued fractions through examples (with numbers),      How would you find the fundamental solution of $x^2 - 95y^2 = 1$ and of $x^2 - 74y^2 = 1$ using continued fractions?
It is quite straightforward to find the fundamental solutions for a given Pell's equation when $d$ is small. But I am unable to solve this equation, as I'm unable to find the fundamental solutions:  Solve: $x^2-29y^2=1$ and $x^2-29y^2=-1$ with $y\not=0$.  Could you please guide me through the solution
Say, for example, you and a group of people were all sailing out in the ocean and something happens...then you say,  "I guess we are all in the same boat"  You are literally in the same boat with everyone else, and your current situation is the same as everyone else.  Now such a phrase can be applied literally and figuratively. I was wondering if there was a word that would mean both literally and figuratively. I know this is a tad bit of an oxymoron, but I was curious to see if there was such an "all encompassing" word for this. Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm? How are the best director award winners chosen? I do not understand how the best director awards are determined each year. It seems impossible to assess the method of a movie director unless you are part of the crew.  So, how do the nominators/judges assess the directors' performance each year? I have Perl code which relies on Term::ReadKey to get the terminal width. My installation is missing this module, so I want to provide a default if the module isn't present rather than throw an exception. How can I conditionally use an optional module, without knowing ahead of time whether it is available. # but only if the module is installed and exists use Term::ReadKey; ...  How can I accomplish this? What's a good phrase for "refining a process which is hopelessly broken"? I'm looking for a turn of phrase to describe a situation where the powers that be wish to continue making small improvements to a process which, due to deep-rooted flaws, will never be close to accurate.  I'm working on a process for estimating the states of business objects, and there's interest in fixing some edge cases which have been noticed.  This could be worthwhile, but because of missing data we're already making heroic guesses.  So if implemented, we might get from 80.0 to 80.1% accuracy.  If we were at 99.5%, I'd be happy to get it to 99.6%, but in the current situation it doesn't seem productive.  We have some CS people on E.SE; what would you call it when someone wants to optimize a two-second process to load data, though once loaded it will be chewed on by an inefficient algorithm for hours?  Something like "bailing the ocean" or "fixing a leaky faucet in a burning building" is close to what I have in mind.  There's also "rearranging deck chairs on the Titanic," but that carries an implication that the changes are wholly useless, and that events are coming to a head. $1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent Firefox - Websites with custom fonts not displaying Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$ I am looking for an English translation of Markov's 1958 paper, On insolubility of the homeorphy problem, which I remember coming across on a website for a computational topology course (taught by Afra Zomorodian). PHP class: Global variable as property in class Showing that $X^2$ and $X^3$ are irreducible but not prime in $K[X^2,X^3]$ Show that $X^2$ and $X^3$ are irreducible but not prime in $K[X^2,X^3]$.   My reasoning is as follows:  Since the only ways $X^2$ and $X^3$ can be factored in $K[X^2,X^3]$ are $X^2 = X^2 * 1$ and $X^3=X^3*1$ it follows that $X^2$ and $X^3$ are irreducible.  To show that they are not prime in $K[X^2,X^3]$ I argue that $X^2\mid X^3*X^3$ but $X^2\nmid X^3$ and $X^3\mid X^4*X^2$ but $X^3\nmid X^4$ or $X^3\nmid X^2$.  Is that correct? I think the single dashes (hyphens) should be placed between the numbers and units in this sentence:     ...in a cylinder of 3-inch radius and 24-inch length.   Is that correct and should hyphens be added to this second sentence?     ...making a trench nearly 15 feet wide and more than 30 feet long. Difference between correlation and covariance: is covariance only useful if the relation is linear? I'm trying to understand better the difference between  covariance and correlation, besides the fact that the correlation coefficient is a dimensional and has values between $-1$ and $1$.   One unclear point is the following: the correlation coefficient can highlight only a linear relation between two variables, but does  covariance only work for linear relations too? Supposing that the relation is not linear, is the covariance not zero? Is it bigger than in the case in which the relation is linear?   Cauchy inequality says $$|\sigma_{xy}|&lt;\sigma_{x} \sigma_{y}$$  Which seems to say that the covariance is maximum when there is a perfect linear relation.    So I do not get what does happen if the relation is not linear, and, if covariance is not a suitable parameter in that cas I don't get the reason for that, since simply looking at the definition:  $$\sigma_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})}{N}$$  it looks like that this should be non zero for any kind of relation (like parabolic).  So can this be a difference between the use of covariance and correlation coefficient? (The first is useful for any kind of relation, the second one only for linear ones). Question Close Updates Pagination not working with custom loop I've got a custom loop that I'm using to display some Real Estate listings that will be available within 60 days. I'm calling it with the following function:   &lt;?php  $sixtydays = date('Y/m/d', strtotime('+60 days')); $paged = (get_query_var('paged')) ? get_query_var('paged') : 1; $query = new PostsOrderedByMetaQuery(array(   'post_type' =&gt; array('post', 'real-estate'),   'meta_key' =&gt; 'Time Available',   'meta_compare' =&gt; '&lt;=',   'meta_value' =&gt; $sixtydays,   'paged' =&gt; $paged,   'orderby_meta_key' =&gt; 'Price',   'orderby_order'    =&gt; 'ASC' )); ?&gt; &lt;?php while ($query-&gt;have_posts()) : $query-&gt;the_post(); ?&gt;   While the loop works great, I can't get it to paginate. It shows the first 10 (my default) posts but doesn't show the pagination. The only way to display all posts is to show them on one page by adding 'posts_per_page' =&gt; -1, I have similar loops on other pages that have no problem paginating. The only difference with this one is that there are two meta keys that are filtering the posts.   I'm using WP Page Navi for this and the rest of my pages. I'm closing the loop and adding the pagination using the following code:   &lt;?php endwhile; // End the loop. Whew. ?&gt; &lt;?php wp_pagenavi(); ?&gt; &lt;?php wp_reset_query(); ?&gt;   How can I go about fixing this? Volumetric Lighting is not working I have a simple setup to try out Volumetric Lighting. I have a spot lamp and Volume Scatter set for the World. The cube has a window for the light of the spot lamp.     But no Volumetric Lighting. I have read most questions on this subject. But I have no idea where I am going wrong. Access user fields? $account = \Drupal::currentUser() loads the user object into $account, but how do I access custom fields I've created on the user object (via the admin UI)?  With dpm($account) I can't see any of my custom fields. Do I need a swap partition? I have 4GB of RAM and I want to install Ubuntu. Do I need a to have a swap partition? Because my computer has 4GB RAM, I do not think it is necessary to have a swap partition. If I do need to have a swap partition, should it be a 4GB or an 8GB partition? Can you win a game of Splendor if you still have cards in reserve? Can my familiar use a Ring of Spell Storing? Can the familiar options from the Pact of the Chain warlock (quasit, imp, sprite, pseudodragon) attune with a ring of spell storing?  If so, can I cast a spell requiring concentration into the ring and then have the familiar maintain concentration on the spell?