Recent Questions - Mathematics Stack Exchange |
- find domains of convergence
- I have a question on Ascoli Arzela theorem in Royden's Book
- Expected Time of no service in a M/M/1 Queue
- When integrating by parts, why does choosing wrong U and V give incorrect answer
- Showing that $\mathcal G$ is a $\sigma-$algebra
- how does EM algorithm work?
- (Sub-)Gaussian Norm Concentration Inequality
- Artin's Example 2.10.3
- The computation of hadamard products of entire functions
- Calculating the derivative of the map $T\to T^{-1}$
- Gelfand-Naimark and the Calkin Algebra
- Distance to the boundary and normal vector
- Find the matricial representation
- Determine the area of the parallelogram that contains the vertices $A = (1,1)$, $B = (2,-1)$ and $C = (1,3)$
- For an $m\times n$ matrix $A, m>n,$ how do I show or know that we can't directly apply the LU decomposition to solve the least squares problem?
- I need help to start this exercise of propositional logic
- How should I check that $6^{\log_{10} x} + 8^{\log_{10} x} = x $ does not have other solutions?
- Specifying trigonometric inequality solutions on an undefined interval - with or without negative angles?
- Is W a subspace of the vector space?
- Generating 2D Triangle and 3D pyramid from list of indices.
- Unknown functions in an integral
- Ask George Casella textbook question 8.37 (a)
- The CPU T, for jobs has gamma distribution mean 40 & standard deviation 20 seconds.. check 3rd question in link please? [closed]
- Geometric Proof of $a^2-b^2=(a+b)(a-b )$ and its applications
- A coin is tossed $10$ times. Find the probability that there exist $7$ consecutive coin tosses with at least $5$ out of the $7$ being heads.
- How to get the LRT statistic of $n(\theta, \sigma^2)$
- Does this product converges in $L^1$?
- Variants of Bernoulli factory problems
- Find standard matrix of linear transformation. Is this right?
- Wieferich's criterion for Fermat's Last Theorem
| Posted: 04 Apr 2021 08:28 PM PDT Let $f_n(x) = \frac{n}{x}e^{-\sqrt\frac{n}{x}} $ Find domain of convergence of the series $\sum_{n=1}^\infty f_n(x) $ Any help??  |
| I have a question on Ascoli Arzela theorem in Royden's Book Posted: 04 Apr 2021 08:26 PM PDT I have a question on Ascoli Arzela theorem. In Royden's Book Theorem is given as follows. Let $\digamma$ be a equicontious family of functions from a separable space $X$ to a metric space $Y$. Let $<f_n>$ be a sequence on $\digamma$ such that for each $x$ in $X$ the closure of the set $\{f_n(x): 0\leq n< \infty \}$ is compact. Then there is a subsequence $<f_{n_k}>$ that converges point wise to a continuous function $f$, and the convergence is uniform on each compact subset of $X$. But when I'm solving problems using Ascoli- Arzela theorem,To show functional space is compact, what I need to show is sequence of functions on compact interval is uniformly bounded on a compact interval and sequence of functions are equicontinuos. I'm confused with the Theorem in Royden's Book. Is it similar to what I have been used in problem solving.  |
| Expected Time of no service in a M/M/1 Queue Posted: 04 Apr 2021 08:26 PM PDT A new cartridge for a ($\lambda_{A}=10,\lambda_{S}=30$)-M/M/1 printer will last for 500 hours of use. What is the expected number of hours before we expect the cartridge to run out? Here, $\lambda_A$ is the arrival rate and $\lambda_S$ is the service rate. I understand the basic concepts in the M/M/1 Queue, but I don't know where to start to calculate the expected time when there are no print jobs. Any hints would be welcome. Please don't reveal the answer, though  |
| When integrating by parts, why does choosing wrong U and V give incorrect answer Posted: 04 Apr 2021 08:23 PM PDT At school we are learning integration by parts and the teacher told us it is important to know what to use as U and V as sometimes it is not so straightforward. If we chose the wrong ones, the working could be a lot more complicated than necessary. However, in the integral $$2n\int_0^1 x^2(1-x^2)^{n-1}dx$$. By letting $u = x^2$ and $v' = (1-x^2)^{n-1}$ I end up with $[2n\frac{x(1-x^2)^n}{-2xn}]_0^1$ + $2n\int_0^1\frac{ (1-x^2)^n}{n}dx $. Since first part is equal to zero, it ends up as $2n\int_0^1 x^2(1-x^2)^{n-1}dx$ = $2\int_0^1(1-x^2)^ndx $ Now using $u = x $ and $v'= x(1-x^2)^{n-1}$ I end up with $[2n\frac{x(1-x^2)^n}{-2n}]_0^1$ + $2n\int_0^1\frac{(1-x^2)^n}{2n}dx$. Since first part equal to zero, it ends up as $2n\int_0^1 x^2(1-x^2)^{n-1}dx$ = $\int_0^1(1-x^2)^ndx$ Now, I know this uses recursive formula and I could've approached it better, but that's not really what I'm stuck on. Why does choosing the U and V as in first case give an incorrect answer. I don't understand why it's wrong, sure the second way is faster, much more logical and etc, but what makes it $\mathit wrong$? What assumption has been made/ rule defied. The teacher told us only to do the second way, but didn't really explain why the first is incorrect. Thanks  |
| Showing that $\mathcal G$ is a $\sigma-$algebra Posted: 04 Apr 2021 08:31 PM PDT I'm new to measure theory and got stucked in the next problem. Given a measure space $(\Omega, \mathcal G, \mu)$ and the set $\mathcal M = \{ X \subset \Omega : \exists A \in \mathcal G \space \text{s.t.} X \subset A, \text{and} \space \mu (A) = 0\}$. Show that $\mathcal G^{\mu} := \{ A = B \cup E, \text{where} B \in \mathcal G \space \text{and} \space E \in \mathcal M\}$ is also a $\sigma-$algebra. So far I've been able to show that $\mathcal G$ is non empty, but haven't been able to show that it is closed under complement and countable unions. For the former, considering an arbitrary $A =B \cup E\in \mathcal G^ \mu$, I tried to show that $A^c = B^c \cap E^c \in \mathcal M$ or $\in \mathcal G$ but didn't get anything. For the latter, no idea yet. Any help is appreciated.  |
| Posted: 04 Apr 2021 08:12 PM PDT I'm working on the Expectation-Maximization recently, and found myself stuck in the process: Question: Consider 6 fair coins {C0,C1,C2,C3,C4,C5 and C6} and the sequential order for the coin toss is as such that C0 is tossed. C1 is tossed only if C0 returns heads, else C2 is tossed. C3 is tossed if C1 returns heads, else C4 is tossed. C5 is tossed if C2 returns heads, else C6 is tossed. The probability of each of the fair coin returning heads is a,b,c,d,e, and f. Can someone explain the expectation-maximization using the Bernoulli distribution?  |
| (Sub-)Gaussian Norm Concentration Inequality Posted: 04 Apr 2021 08:09 PM PDT I am looking for a proof of the following: Let $X_1 \dots, X_n $ be i.i.d. $N(\mu, \Sigma)$, then $$ \mathbb{P} \left ( \|\bar{X}_n-\mu \|_{2} > \sqrt{\frac{\text{Tr}\Sigma}{n}}+\sqrt{\frac{2 \| \Sigma \| \log(1/\delta)}{n}} \right ) \le \delta $$ where $\|\Sigma \|$ is the largest eigenvalue of $\Sigma$. This holds generally for any sub-Gaussian random variables. I am not sure how to get the $\| \Sigma\|$ term - I'm just looking for a hint or a good reference here.  |
| Posted: 04 Apr 2021 08:17 PM PDT In example 2.10.3, if H is a subgroup (odd order) of a Symmetric group, then H is contained in the kernel i.e H is a subgroup of Alternating group. But Artin says H is a cyclic subgroup, can anyone explain?  |
| The computation of hadamard products of entire functions Posted: 04 Apr 2021 08:06 PM PDT The Hadamard factorization theorem confirms the existence of Hadamard products of entire functions. But is there a feasible computational technique to find the Hadamard product of a given entire function? My question originates from an exercise on Stein's Complex Analysis asking me to find Hadamard products of $e^{z} - 1$ and $\cos\pi z$. I found a question here, and it seems that the answer uses the characteristics of the functions themselves to get a hadamard product, and the method is not general for all entire functions.  |
| Calculating the derivative of the map $T\to T^{-1}$ Posted: 04 Apr 2021 08:21 PM PDT Let $E$ be a banach space and $U$ be the set of all bounded invertible linear operator on $E$ with bounded inverse. Consider the map, $f:U\to U$ by $T\to T^{-1}$ Now we are interested in calculating the 2nd order derivatives of this map. For the first order derivative, I got that $D(f(T))(S)=-S^{-1}TS^{-1}=-M(T),$ where $M:BL(E)\to BL(E)$ given by, $M(T)=S^{-1}TS^{-1}$ Let, $A: BL(E)\to BL(E)$ denoted by, $A(T)=T(S)$ be the evaluation map at $S.$ So , we have, $ A(D(f(T)))=-M(T).$ Therefore by chain rule, we get, $$D^2(f(T))(S,W)=-DM(T)(W)=-S^{-1}WS^{-1}.$$ It seems I am doing something wrong. Can anyone point it out?  |
| Gelfand-Naimark and the Calkin Algebra Posted: 04 Apr 2021 08:01 PM PDT On https://en.wikipedia.org/wiki/Calkin_algebra, "The Gelfand-Naimark-Segal construction implies that the Calkin algebra is isomorphic to an algebra of operators on a nonseparable Hilbert space." Let $H$ be an infinite dimensional, complex, separable Hilbert space. By applying the Gelfand–Naimark-Segal Construction we can obtain an injective *-homomorphism $\phi: B(H)/K(H) \rightarrow B(H')$ for some Hilbert space $H'$, where $B(H)/K(H)$ is the Calkin algebra. I want to show that $H'$ here is not separable in this case. Let's try proving it by contradiction. Suppose there exists a countable dense subset of $H'$. And let $\{ e_n\}_{n\in \mathbb{Q}}$ be an orthonormal basis for $H$. I know that there exists uncountable many infinite subsets $\{E_{\alpha}\}_{\alpha \in I}$ of $\mathbb{Q}$ such that $E_{\alpha } \cap E_{\beta}$ is finite for $\alpha \ne \beta$. I was suggested to consider the orthogonal projections $P_{\alpha}$ of $H$ onto span $\{e_n: n \in E_{\alpha}\}$... But I'm not sure how to proceed from here. Any references or suggestions will be appreciated! Thank you!  |
| Distance to the boundary and normal vector Posted: 04 Apr 2021 07:56 PM PDT Let $\Omega\subset \mathbb{R}^n$ be open, bounded with smooth boundary ($C^2$ at least). Let $x_0\in \partial\Omega$, we know that the outward normal vector is given by $$ \mathbf{n}(x_0) = -\nabla d(x_0)$$ where $d(x) = \mathrm{dist}(x,\partial \Omega)$ is the distance function. My question is, how to show that for $s$ small there holds $$ d(x_0-s\mathbf{n}(x_0)) = s?$$  |
| Find the matricial representation Posted: 04 Apr 2021 08:10 PM PDT Let $M_{2}\mathbb{R}$ be the matrix space of $2 \times 2$ with real entries , and $P_{2}[x]$ the space of polinomies with real coeficientes with grade at most 2. Considerate the linear transformation $T: M_{2}(\mathbb{R}) \rightarrow P_{2}[x]$ defined like $$T \begin{pmatrix} a & b \\ c & d \end{pmatrix}= (a+b)+2dx+bx^{2}$$ Calculate the matricial representation of $T$ respect the canonic base of $M_{2}(\mathbb{R})$ and $P_{2}[x]$ respectively: $$B_{1}= \{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \} $$ $$B_{2}= \{1,x,x^{2} \} $$ I got $T(e_{1})= 1$, $T(e_{2})=1+x^{2}$, $T(e_{3})=0$, $T(e_{4})=2x$, so $$T= \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2\\ 0 & 1 &0 & 0 \end{pmatrix}$$ Am I right?  |
| Posted: 04 Apr 2021 08:28 PM PDT I don't understand how to solve any examples like this questions. Can someone explain answering this one for me and giving more examples?  |
| Posted: 04 Apr 2021 07:57 PM PDT
I've seen another post that discusses something sort of surrounding it, but doesn't quite reach what I'm wondering. My intuition is that using the LU decomposition to find the LS solution is ill-conditioned, since we use the QR decomposition instead of the normal equations because despite the cost of the QR algorithm the normal equations are far worse conditioned. Am I in the right direction?  |
| I need help to start this exercise of propositional logic Posted: 04 Apr 2021 08:27 PM PDT I have the definition of this function \begin{equation*} \begin{aligned} Subf : PROP & \rightarrow \mathcal{P}(PROP) \\ F & \mapsto Subf(F) \end{aligned} \end{equation*}with, $Subf(F) = \{F\}$, $Subf(\neg F) = \{ \neg F\} \cup Subf(F)$ and $Subf(F \circ G) = \{F \circ G\} \cup Subf(F) \cup Subf(G)$ this function determines the set of sub-formulas of a given a propositional formula and i must show that if $F$ has $n$ connectives then $Subf(F)$ has at most $2n + 1$ propositional formulas. I have no idea that how to start proof it. Any ideas?. Thank you  |
| How should I check that $6^{\log_{10} x} + 8^{\log_{10} x} = x $ does not have other solutions? Posted: 04 Apr 2021 07:56 PM PDT The question is to solve the equation $6^{\log_{10} x} + 8^{\log_{10} x} = x $ I know one of the solutions is $x=100$ using Pythagorean triples but I can't show that this is the only solution. I'm looking for an idea without using differentiation! any hints would be appreciated. tnx  |
| Posted: 04 Apr 2021 08:00 PM PDT Suppose I'm trying solve the following inequality for $x$, on an unspecified interval: $$\cos2x \ge -\tfrac{1}{2}$$ An obvious solution would be $x \in [-\frac{\pi}{3} + k\pi, \frac{\pi}{3} + k\pi]$ - (where $k \in \mathbb{Z}$). What I'm wondering is if the solution could be written such that it omits the initial negative angle, for example: $x \in [k\pi, \frac{\pi}{3} + k\pi] \cup [\frac{2\pi}{3} + k\pi, \pi + k\pi]$. Both solutions cover the same region of the unit circle (sorry if my terminology isn't good here), however only the first one includes a negative solution (in the lower bound). I'm wondering whether both solutions are correct/valid.  |
| Is W a subspace of the vector space? Posted: 04 Apr 2021 08:21 PM PDT Question: W is the set of all vectors in R3 whose first component is -3. Is W a subspace of the vector space? If not, state why. My Answer: W is not a subspace of the vector space because it is not closed under addition and neither is it closed under scalar multiplication. But how do I provide a counterexample to this question? I'm unsure of how to do it. Please help, thank you.  |
| Generating 2D Triangle and 3D pyramid from list of indices. Posted: 04 Apr 2021 08:26 PM PDT Given
How would I use the numbers' value to generate a list of points the create a triangle or pyramid? I know to make a grid, I could do: So I think for the x value at least, I would maybe still mod i, but it would by by some function of i that would return the size of the given row it is on: I subtract (f(i) / 2) to keep the rows centered. Then similarly the y value, would need to increase by 1 I think the same way Example:  |
| Unknown functions in an integral Posted: 04 Apr 2021 08:10 PM PDT I'm having a problem with an integral that I do not understand. The form of my equation is $2Kp(x)/dx = -p(x)g$. I need to integrate this, but I'm not sure how to integrate it. Assuming I move the dx to the right side of the equation and integrate will it be $(-p(x)gx) - ((-p(x)^2/2)g)$?  |
| Ask George Casella textbook question 8.37 (a) Posted: 04 Apr 2021 08:08 PM PDT This is from George Casella textbook question 8.37 (a). Let $X_1,...,X_n$ be a random sample from a $n(\theta, \sigma^2)$ population. Consider testing $H_0:\theta\leq \theta_0$ versus $H_1:\theta> \theta_0$. If $\sigma^2$ is known, show that the test that rejects $H_0$ when $$\bar{X}>\theta_0+z_\alpha \sqrt{\sigma^2/n}$$ is a test of size $\alpha$. Show that the test can be derived as an LRT. My understanding of this question is it contains 2 questions. First is to show the size of this test is $\alpha$. The second question is to show this test can be derived as an LRT. I have both problems on these 2 questions. My attempt of the first question: to show the size is $\alpha$. I show it based on the definition strictly. According to the definition 8.3.5, $\alpha=sup\beta(\theta)=supP(x\in R|H_0)=supP(x\in R|\theta\leq \theta_0)$. Then I first get $P(x\in R)=P(\bar{X}>\theta_0+z_\alpha \sqrt{\sigma^2/n})=P(\bar{X}-\theta>\theta_0-\theta+z_\alpha \sqrt{\sigma^2/n})=P((\bar{X}-\theta)/\sqrt{\sigma^2/n}>(\theta_0-\theta)/\sqrt{\sigma^2/n}+z_\alpha)=P(Z>(\theta_0-\theta)/\sqrt{\sigma^2/n}+z_\alpha)$. Then I find this probability is increasing in terms of $\theta$, so the supremum under $H_0=\theta\leq \theta_0$ is reached at $\theta=\theta_0$. So the above becomes $P(Z>z_\alpha)$. It is exactly $\alpha$. Then I finish my proof to show the size of this test is $\alpha$. But the solution is very easy. I think the solution is wrong. Because it is not $Z=(\bar{X}-\theta_0)/\sqrt{\sigma^2/n}$, it is $Z=(\bar{X}-\theta)/\sqrt{\sigma^2/n}$. So I think the solution is wrong. But 8.37(a), 8.37(c) both use this solution. Am I correct? For the second question: to show this test can be derived as an LRT. Yes, I understand and get the step in the solution: it is equivalent to rejecting if $(\bar{x}-\theta_0)/\sqrt{\sigma^2/n}>c'$. Then we finished the proof? Shouldn't we derive the exact same form of $\bar{X}>\theta_0+z_\alpha \sqrt{\sigma^2/n}$? The solution is:  |
| Posted: 04 Apr 2021 07:55 PM PDT The CPU T, for jobs has gamma distribution mean 40 & standard deviation 20 seconds. Any job taking less than 20 seconds is a short job.What the probability that of 5 randomly selected jobs at least 2 jobs are short? X20.193/4 X0.379/4 X3.0.855/4?  |
| Geometric Proof of $a^2-b^2=(a+b)(a-b )$ and its applications Posted: 04 Apr 2021 08:26 PM PDT I am teaching a student on the subject of factoring. One commonly used formula is $a^2-b^2=(a+b)(a-b) $. It is easy to prove it from RHS to LHS algebraically, but how to prove it geometrically? I would also like to find some of its applications, such as this:$(\sin\theta)^2+(\cos\theta)^2=( \cos \theta +i\sin \theta)(\cos\theta -i\sin \theta )=e^{i\theta}\cdot e^{-i\theta } =e^0=1$ Are there any other good examples? EDIT: some comments for applications (Albus) $\frac{1}{\sqrt a+b}=\frac{\sqrt a-b}{a-b^2}$ (Paul) $a\times b=(\frac{a+b}{2})^2-(\frac{a-b}{2})^2$  |
| Posted: 04 Apr 2021 08:17 PM PDT
So for example, $TTHHTHHTHH$ is one of the outcomes we want. I guess the best way to treat this is as a counting problem; finding the number of outcomes we want and then dividing by $2^{10}.$ $2^{10} = 1024,$ so listing all the outcomes is time-wise expensive. The events, "The first consecutive $7$ tosses contain at least $5$ heads", "The second consecutive $7$ tosses contain at least $5$ heads", etc. are not mutually exclusive and so the answer is not simply: $P$(The first consecutive $7$ tosses contain at least $5$ heads) + $P$(The second consecutive $7$ tosses contain at least $5$ heads) + ... . Similarly, the events, "The first consecutive $7$ tosses does not contain at least $5$ heads", "The second consecutive $7$ tosses does not contain at least $5$ heads", etc. are also not mutually exclusive and so the answer is not simply: $1 - $ $[P$(The first consecutive $7$ tosses does not contain at least $5$ heads) + $P$(The second consecutive $7$ tosses does not contain at least $5$ heads) + ...$]$ . But I also don't see a clear way to get round this double counting via inclusion-exclusion principle. Although I feel like I'm bad with inclusion-exclusion in general tbh. Edit: What about reflecting the $10$ boxes down the middle? This could cut our work in half maybe? For example, $HTTHHTHTHH \equiv HHTHTHHTTH$. I'm also interested in doing a similar problem with larger numbers, e.g.:
This is probably impractical to calculate using binomial distributions, so how would you find an answer using the Normal distribution as an approximation, or is it not possible to do this? The reason I'm interested in this latter question is that it is the sort of calculation one might make if one wanted to gain statistical evidence that a poker site is rigged against them, although of course the latter question would not be enough evidence to prove a poker site is rigged against a particular player; it could be a reasonable starting point for further calculations. Also, it is not hard to imagine this calculation could have applications in other areas, statistical mechanics or mathematical biology for example.  |
| How to get the LRT statistic of $n(\theta, \sigma^2)$ Posted: 04 Apr 2021 07:56 PM PDT This is from George Casella textbook question 8.37 (c). Let $X_1,...,X_n$ be a random sample from a $n(\theta, \sigma^2)$ population. Consider testing $H_0:\theta\leq \theta_0$ versus $H_1:\theta> \theta_0$. If $\sigma^2$ is unknown, what is the LRT statistic? The solution is $\lambda=(\hat{\sigma}^2/\hat{\sigma_0}^2)^{n/2}$. I didn't get this. My attempt: the LRT of denimoator is easier and reached when $\theta=\bar{x}$ and $\sigma^2=\hat{\sigma}^2$ below in the solution. The numerator is $\sigma^2=\hat{\sigma}^2$ and $\theta=\theta_0$.(We don't need to consider when $\lambda=1$ case.) I didn't get the LRT statistic in the solution.
 |
| Does this product converges in $L^1$? Posted: 04 Apr 2021 08:23 PM PDT Suppose $(X_n)$ is a discrete sequence of independent and equally distributed random variables such that : $E[X_3]=1$ and $P(X_3=\alpha)$ +$P(X_3=\beta)=1$. Does the product $P_n=\Pi_{k=1}^n X_n$ converges in $L^1$? my attempt is that : $E[P_n]=(E[X]^n)=1$ so it doesn't converge but I'm not sure Edit: I forgot to add that $\alpha$ is in $]0,1[$ and $\beta \gt \alpha$  |
| Variants of Bernoulli factory problems Posted: 04 Apr 2021 08:08 PM PDT I am interested in the following two questions related to the famous Bernoulli factory problems.
Any reference/idea will be much appreciated for the multivariate case, thanks!  |
| Find standard matrix of linear transformation. Is this right? Posted: 04 Apr 2021 08:01 PM PDT I am a tad confused about a couple of problems:
Is this as simple as: $$\begin{bmatrix} 1 & -2 & 1 \\ 0 & 1 & -2 \\ 1 & 0 & 3 \end{bmatrix}$$ The theorem I'm relying on is this:
In this proof, how does one get from the column of a's to the $T(ei)$ at the end?
a) $$\begin{bmatrix} 1 & -1 & 1 \end{bmatrix} = A$$ b) $A * (1,-2,2) = (1,2,2)$ Is this right?  |
| Wieferich's criterion for Fermat's Last Theorem Posted: 04 Apr 2021 08:11 PM PDT I have found the following way to prove some(Wieferich's) criterion for Fermat's Last Theorem and am wondering what would be wrong. My point of doubt is calculation of the Fermat-quotients of $y,z$ being $-1$, since I found these rules on Wikipedia. Also, should I split this in parts? I can imagine people don't feel like going through too much text. Anyway, have fun! Theorem: Let: $ \quad \quad \quad \quad p$ be an odd prime, $ \quad \quad \quad \quad \gcd(x,y,z) = 1$, $ \quad \quad \quad \quad xyz \not \equiv 0 \pmod p$ If: $ \quad \quad \quad \quad x^p = y^p + z^p$, then $p$ is Wieferich-prime. Proof: Consider the following congruence: $ \quad \quad \quad \quad (x^n - y^n)/(x - y) \equiv nx^{n - 1} \pmod {x - y}$ which we can prove by induction on $n$ and in which we divide first. Let $n = p$. Since: $ \quad \quad \quad \quad \gcd(x - y,(x^p - y^p)/(x - y))$ $ \quad \quad \quad \quad = \gcd(x - y,px^{p - 1})$ $ \quad \quad \quad \quad = \gcd(x - y,p)$ $ \quad \quad \quad \quad = \gcd(x - z,p)$ $ \quad \quad \quad \quad = 1$, it follows that: $ \quad \quad \quad \quad x - y = r^p$, $ \quad \quad \quad \quad (x^p - y^p)/(x - y) = s^p$, $ \quad \quad \quad \quad x - z = t^p$, $ \quad \quad \quad \quad (x^p - z^p)/(x - z) = u^p$, $ \quad \quad \quad \quad rs = z$, $ \quad \quad \quad \quad tu = y$, for some $r,s,t,u$ with $\gcd(r,s) = \gcd(t,u) = 1$. The following also holds for $x - z,t,u$: $ \quad \quad \quad \quad s \equiv 1 \pmod p \implies s^p \equiv 1 \pmod {p^2}$ Now let: $ \quad \quad \quad \quad s^p = px^{p - 1} \pmod {x - y}$ $ \quad \quad \quad \quad \implies s^p = px^{p - 1} + ar^p \equiv 1 \pmod {p^2}$, for some $a$ $ \quad \quad \quad \quad \implies s \equiv ar \equiv 1 \pmod p \implies s^p \equiv (ar)^p \equiv 1 \pmod {p^2}$ $ \quad \quad \quad \quad \implies ar^p \equiv 1/a^{p - 1} \pmod {p^2}$ $ \quad \quad \quad \quad \implies s^p = px^{p - 1} + ar^p \equiv px^{p - 1} + 1/a^{p - 1} \equiv 1 \pmod {p^2}$ $ \quad \quad \quad \quad \implies px^{p - 1} \equiv 1 - 1/a^{p - 1} \pmod {p^2}$ $ \quad \quad \quad \quad \implies p(ax)^{p - 1} \equiv a^{p - 1} - 1 \pmod {p^2}$ $ \quad \quad \quad \quad \implies q_p(a) \equiv 1 \pmod p$, where $q_p(a)$ denotes the Fermat-quotient for $a$ modulo $p$. So it follows that: $ \quad \quad \quad \quad q_p(r) \equiv q_p(1/a) \equiv -q_p(a) \equiv -1 \pmod p$ Because of $q_p(s) \equiv 0 \pmod p$: $ \quad \quad \quad \quad q_p(z) \equiv q_p(rs) \equiv q_p(r) + q_p(s) \equiv -1 + 0 \equiv -1 \pmod p$ Since the same holds for $x - z,t,u$, we now have: $ \quad \quad \quad \quad q_p(y) \equiv q_p(z) \equiv -1 \pmod p$ From which it follows that: $ \quad \quad \quad \quad y^{p - 1} \equiv 1 - p \pmod {p^2}$ $ \quad \quad \quad \quad z^{p - 1} \equiv 1 - p \pmod {p^2}$ $ \quad \quad \quad \quad \implies y^p \equiv y(1 - p) \pmod { p^2 }$ $ \quad \quad \quad \quad \implies z^p \equiv z(1 - p) \pmod { p^2 }$ We also note: $ \quad \quad \quad \quad y^p \equiv (tu)^p \equiv t^p \equiv x - z \pmod {p^2}$ $ \quad \quad \quad \quad z^p \equiv (rs)^p \equiv r^p \equiv x - y \pmod {p^2}$ So we can set: $ \quad \quad \quad \quad y(1 - p) \equiv x - z \pmod {p^2}$ $ \quad \quad \quad \quad z(1 - p) \equiv x - y \pmod {p^2}$ $ \quad \quad \quad \quad \implies (x - z)/y \equiv (x - y)/z \implies z(x - z) \equiv y(x - y) \pmod {p^2}$ $ \quad \quad \quad \quad \implies y^2 - z^2 \equiv (y + z)(y - z) \equiv x(y - z) \pmod {p^2}$ So either: $ \quad \quad \quad \quad \implies x \equiv y + z \pmod {p^2}$ or: $ \quad \quad \quad \quad \implies p | y - z$ Suppose $x \equiv y + z \pmod {p^2}$: $ \quad \quad \quad \quad \implies (x - y)^p \equiv r^{p^2} \equiv r^p \equiv x - y \implies z^p \equiv z \pmod {p^2}$, contradicting: $ \quad \quad \quad \quad z^p \equiv z(1 - p) \pmod { p^2 }$ So now we know $y \equiv z \pmod {p}$ But then: $ \quad \quad \quad \quad y^p \equiv z^p \pmod {p^2}$ $ \quad \quad \quad \quad \implies x^p \equiv y^p + z^p \equiv 2z^p \pmod {p^2}$ Also: $ \quad \quad \quad \quad x \equiv y + z \implies x \equiv z + z \equiv 2z \pmod p$ $ \quad \quad \quad \quad \implies x^p \equiv (2z)^p \pmod {p^2}$ We conclude: $ \quad \quad \quad \quad x^p \equiv (2z)^p \equiv 2z^p \pmod {p^2}$ $ \quad \quad \quad \quad \implies (2z)^p - 2z^p \equiv 0 \pmod {p^2}$ $ \quad \quad \quad \quad \implies z^p(2^p - 2) \equiv 0 \pmod {p^2}$, from which we can see $p$ must be a Wieferich-prime.  |
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