A few days ago, I told my friend that I wanted to learn trigonometry by the end of $2018$. His immediate reply was that I should go on if I could solve this mathematics problem:      a and b, given that $a\text{ and } b \in \mathbb N$ and $\left( \left( a^2+b^2 \right) \div \left( ab+1 \right)\right) \in \mathbb N$, then prove that $\sqrt{\frac {\left( a^2+b^2 \right) }{ \left( ab+1 \right)}} \in \mathbb N$   Over the last few days, I went about trying to see if I could prove it. (My friend has a tendency to occasionally trick me with unsolvable math problems) this is what I proved:   If $a \text{ xor } b=0$, it works. If $b=a^3 \text{ and } a \neq 1$, it works. Besides $n=0$, $x^n=a \text{ and } x^{n+2}=b $will work   To recap, one number must be $2$ powers greater than the other (i.e. $32$ and $128$. [$2^5$ and $2^7$]).    Or, at least what I thought...    When I sent an email to him, he said that I missed 2 very seldom noticed things and that everyone he had ever asked the problem missed it.  Which either means he intentionally made the problem too hard for my intelligence ( which is -spoiler alert- very low, compared to you people), or he is just making fun of me for not noticing his problem is impossible. Either way, a plethora of mathematicial geniuses whom are much more experienced than me at algebra could solve this easily. Could someone tell me what I'm missing??  Thanks in advance.   (And DON'T tell my friend about this!!)
Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English.
What are examples of two non-similar invertible matrices with same minimal and characteristic polynomial and same dimension of each eigenspace? I have purchased a rather large bag of long-grain white rice (25 lbs.) and need to know the best way for storing it long-term. This rice will be used as both a food-storage food source and as my everyday source of rice for cooking, so the storage solution should (hopefully) be resealable, rather than once-it's-open-it's-open. I eat rice maybe three times a month, so it isn't really "everyday" but it will be my go-to when I have it. Does a (half) beheaded Ettin take bleeding damage of some sort? Let's say a character cuts off one of an Ettin's heads, but the other head is still intact, and it's as such still alive. Obviously, there's a lot of blood. Does the Ettin take continuous damage from naturally bleeding out? Or does it not work that way? 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I have a particular situation here, In March of 1997 I travelled to the US as a tourist and lived there illegally for almost 9 years. I was never arrested, or had any trouble with the law, I was never deported. I paid my taxed (had a TFN) had a job, bank account bought a house, etc. About 9 years ago I decided to leave as I was tired of living like that, not able to be legal, in fear of deportation and loosing everything. I also got married (to a person in the exact situation as me) we were able to legally migrate to Australia and now we are Australian citizens. We never returned our I-94s and in 2005 left through Mexico (walked through the border) so there are no records of us ever leaving. We have lived here in Australia for over 9 years.  OK so my question is... We have been thinking we would like to visit our friends and family back in the US now that we have a little girl. Is there any chance they won't let us in because they know we overstayed? 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QR Codes generation from MySQL rows When to use (and not use) the rule of three How do you do a deep copy of an object in .NET? find all $(x,y,z)$ such that $27^{3x^2 + 2y}+27^{3y^2 + 2z}+27^{3z^2 + 2x}=1$ Find all ($x,y,z$) such that $$ 27^{3x^2 + 2y}+27^{3y^2 + 2z}+27^{3z^2 + 2x}=1$$  I am a high school student and would appreciate it if anyone could solve it using high school mathematics. here is what I have tried till now: $$ 3x^2 +2y &lt; 0 $$ (this can also be said for the other 3 equation. it also means that x,y,z&lt;0) $$ -2y&gt;3x^2 $$ $$ 4y^2&gt;9x^2 $$ ( -2y and 3x^2 are both positive) $$ 3y^2&gt;27/4 x^2 $$ $$ -2z&gt;3y^2&gt;27/4 x^4 $$ $$ 4z^2 &gt; 729/16 x^8 $$ $$ 3z^2&gt; 2187/64 x^8 $$ $$ -128/2187&lt;x^7 $$ $$ -2/3 &lt; x $$ $$ -2/3&lt; x,y,z &lt; 0 $$ i tried this but as you can see didn't do much good Replace non-numeric with empty string Quick add on requirement in our project.  A field in our DB to hold a phone number is set to only allow 10 characters.  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