## The Legendre Symbol

Prove

$latex x^{2} \equiv 3411 \mod 3457$
has no solution?

Legendre Symbol:

$latex \displaystyle x^{2} \equiv a \mod p \iff \boxed{ \left( \frac {a}{p} \right) = \begin{cases} -1, & \text{if 0 solution} \\ 0 , & \text{if 1 solution} \\ 1, & \text{if 2 solutions} \\ \end{cases} }$

Hint: prove $latex \left( \frac{3411}{3457} \right) = -1$

Using the Law of Quadratic Reciprocity, without computations, we can prove there is no solution for this equation.

Solution:

1.
3411 = 3 x 3 x 379 = 9 x 379

$Latex \displaystyle \boxed{ \left(\frac{a}{p}\right) \left(\frac{b}{p} \right)= \left(\frac{ab}{p}\right) }$

$latex \displaystyle \left(\frac{3411}{3457} \right)= \left(\frac{9}{3457} \right).\left(\frac{379}{3457} \right)= \left(\frac{379}{3457} \right)$
since
$latex \displaystyle\left(\frac{9}{3457} \right)=1$
because 9 is a perfect square, 3457 is prime.

$latex \displaystyle \boxed{ \text{If p or q or both are } \equiv 1 \mod 4 \implies \left(\frac{p}{q} \right)= \left(\frac{q}{p} \right)}$

Since
$latex… View original post 212 more words ## Математика Групповые занятия класса, чтобы начать в следующем году, 2014 году. # Математика Групповые занятия класса, чтобы начать в следующем году, 2014 году.Математика Обучение центр ## The Singapore Math The famous Singapore Math for children in primary schools is based on visual models. The Singapore Ministry of Education has published a new 2013 Math syllabus for primary and secondary schools, which will roll out in examinations within 4 to 6 years. Todate only Primary 1 and Secondary 1 Math syllabuses are published here: http://www.moe.gov.sg/education/syllabuses/sciences View original post ## Algebra vs Singapore Math Who wins? This comic video illustrates Singapore Math’s Arithmetics Polya-style problem solving process vs Algebra’s mechanical method. The problem is as follow: R is 3 times older than S two years ago. From now 2 years later, their total age is 32. How old is R now ? See my previous blog (search “Monkey”) the Nobel Physicist Paul Dirac’s problem “The Monkeys and Coconuts“, 3 methods are used: 2 adanced modern math (by Sequence, eigenvector & eigenvalue), and the easiest & intuitive method (by Singapore Modelling Math). High-school Algebra method is impossible, if not cumbersome, to solve the Monkey problem ! View original post ## Khan Academy I find Khan Linear Algebra video excellent. The founder / teacher Sal Khan has the genius to explain this not-so-easy topic in modular videos steps by steps, from 2-dimensional vectors to 3-dimensional, working with you by hand to compute eigenvalues and eigenvectors, and show you what they mean in graphic views. 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Inverse of log (bijective):$latex \log \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= \log 2latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$View original post ## Prof Su Buqing Problem Prof Su 苏步青, the founding pioneer Math professor of the China’s top universities (Zhejiang 浙江大学 and Fudan 复旦大学), was one of the few mathematicians who had longevity above 100 years old (the other was French Mathematician Hadammard). http://en.m.wikipedia.org/wiki/Su_Buqing Two men A and B are 100 km apart, walking towards each other, A at speed 6 km/hour and B at 4 km/hour. A brings a dog which runs at 10 km/hour between them, starting from A towards B, upon reaching B it runs back to reach A, then back to B again, and so on… Find total distance the dog has covered when A and B finally meet ? 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They have a certain amount and that’s that; nothing can be done to change it. In the growth mindset, people believe that their talents and abilities can be developed through passion, education, and persistence.” Jo states that the fixed mindset contributes to one of the biggest myths in mathematics: being good at math is a gift. She referenced her book, The Elephant in the Classroom (added it to my reading list) and showed the audience various television/movie clips that continue to perpetuate… View original post 760 more words ## Checking Multiplication via Digit Sums Last week a friend who is a fourth grade teacher came to me with a math problem. The father of one of his students had showed him a trick for checking the result of a three-digit multiplication problem. The father had learned the trick as a student himself, but he didn’t know why it worked. My friend showed me the trick and asked if I had seen it before. This post describes this check and explains why it works. 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The short version of my opinions is that multiple choice quizzes have significant limitations when used in the traditional classroom setting, but have a lot of interesting and underexplored potential when used as a self-assessment tool. View original post 2,490 more words ## Why aren’t all functions well-defined? I’m in the happy state of just having finished marking exams for this year. There is very little of interest to say about the week that was removed from my life: it would be fun to talk about particularly bizarre mistakes, but I can’t really do that, especially as the results are not yet known (or even fully decided). However, one general theme emerged that made no difference to anybody’s marks. There seems to be a common misconception amongst many Cambridge undergraduates that I’d like to discuss here in the hope that I can clear things up for a few people. 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I’m leaving the previous thread open for those who wish to respond directly to some specific comments in that thread, but otherwise it would be preferable to start afresh on this thread to make it easier to follow the discussion. It’s not easy to summarise the discussion so far, but the comments have identified several existing formats for displaying (and marking up) mathematics on the web (mathMLjsMath, MathJaxOpenMath), as well as a surprisingly large number of tools for converting mathematics into web friendly formats (e.g. LaTeX2HTMLLaTeXMathML, LaTeX2WPWindows 7 Math Inputitex2MMLRitexGellmumathTeXWP-LaTeXTeX4htblahtexplastexTtHWebEQtechexplorer View original post 325 more words ## Gamifying algebra? High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. 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One might… View original post 9,949 more words ## Quiz In the diagram, the circumference of the external large circle is 1) longer, or 2) shorter, or 3) equal to, the sum of the circumferences of all inner circles centered on the common diameter, tangent to each other. Answer: 3) equal circumference = π. diameter Let d be the diameter of the external large circle C Let dj be the diameter of the inner circle Cj$latex \displaystyle d = \sum_{j} d_jlatex \displaystyle \pi. d = \pi. \sum_{j} d_j= \sum_{j}\pi.d_j$Circumference of the external circle = sum of circumferences of all inner circles View original post ## What maths A-level doesn’t necessarily give you I had a mathematical conversation yesterday with a 17-year-old boy who is in his second year of doing maths A-level. 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Example:$latex \sqrt[3]{658503} = N$Last three digits 503 <-> …[7] First three digits 658: (8³ =512)< 658 < (729 = 9³) => 8 Answer :$latex \sqrt[3]{658503} = N$= 87 Note: Similar trick for opening$latex \sqrt[23] {200 digits}$by an indian lady Ms Shakuntala (83) dubbed “Human computer”. View original post ## Solution 2 (Eigenvalue): Monkeys & Coconuts Solution 2: Use Linear Algebra Eigenvalue equation: A.X = λ.X A =S(x)=$Latex \frac{4}{5}(x-1)$where x = coconuts S(x)=λx Since each iteration of the transformation caused the coconut status ‘unchanged’, which means λ = 1 (see remark below)$Latex \frac{4}{5}(x-1)=x$We get x = – 4 Also by recursive, after the fifth monkey:$Latex S^5 (x)$=$Latex (\frac{4}{5})^5 (x-1)- (\frac{4}{5})^4-(\frac{4}{5})^3- (\frac{4}{5})^2- \frac{4}{5}Latex S^5 (x)$=$Latex (\frac{4}{5})^5 (x) – (\frac{4}{5})^5 – (\frac{4}{5})^4 – (\frac{4}{5})^3+(\frac{4}{5})^2 – \frac{4}{5}Latex 5^5$divides (x) Minimum positive x= – 4 mod ($Latex 5^{5}$)=$Latex 5^{5} – 4$= 3,121 [QED] Note: The meaning of eigenvalue λ in linear transformation is the change by a scalar of λ factor (lengthening or shortening by λ) after the transformation. Here λ = 1 because “before” and “after” (transformation A) is the SAME status (“divide coconuts by 5 and left 1”). 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x = \frac{-b \pm \sqrt{b^{2}-4ac}}
{2a}
}$2.$latex \mathbb{NZQRC}$Nine Zulu Queens Rule China 3.$latex \boxed {\cos 3A = 4\cos^{3}…

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## What is “sin A”

What is “sin A” concretely ?

1. Draw a circle (diameter 1)
2. Connect any 3 points on the circle to form a triangle of angles A, B, C.
3. The length of sides opposite A, B, C are sin A, sin B, sin C, respectively.

Proof:
By Sine Rule:

$latex \frac{a}{sin A} = \frac{b}{sin B} =\frac{c}{sin C} = 2R = 1$
where sides a,b,c opposite angles A, B, C respectively.
a = sin A
b = sin B
c = sin C

View original post