Firstly, apologies for the long gap.  Very far from being Theorem of the Week, I know.  Here’s another theorem for now, and I’ll do what I can to revert to a weekly post.

So, to this week’s theorem.  I have previously promised to write about Fermat‘s Little Theorem, and I think it’s time to keep that promise.  In that post (Theorem 10, about Lagrange’s theorem in group theory), I introduced the theorem, so I’m going to state it straightaway.  If you haven’t seen the statement before, I suggest you look back at that post to see an example.

Theorem (Fermat’s Little Theorem) Let $latex p$ be a prime, and let $latex a$ be an integer not divisible by $latex p$.  Then $latex a^{p-1} \equiv 1\mod{p}$.

If you aren’t comfortable with the notation of modular arithmetic, you might like to phrase the conclusion of the theorem as saying that $latex…

Excellent Notes on Group Theory.

http://lishen.files.wordpress.com/2012/07/groups6.pdf

Senior Wrangler is the First position in the Math Tripos in Cambridge. Singapore Prime Minister Lee Hsien Loong was the Senior Wrangler in 1973, the first Singaporean student with such great honors, among other senior wranglers like Arthur Cayley (Group Theory), J.J. Sylvester (Inventor of Matrix, private tuitor of the “inventor of Nursing” Florence Nightingale), J.E. Littlewood (partnered in a twin research team with G.H. Hardy), Frank Ramsey (Ramsey’s Theorem), Stokes, Pell, etc.

Some great mathematicians like Bertrand Russell (Logician, Nobel Litterature Prize) , G.H. Hardy (20th century greatest Pure Mathematician, mentored 2 geniuses: Indian Ramanujian and Chinese Hua Luogeng 华罗庚*) were not Senior Wrangler. Prof Hardy hated Math Tripos syllabus (revealed in his autobiography: “A Mathematician’s Apology“).

1914 Brian Charles Molony
1923 Frank Ramsey
1928 Donald Coxeter
1930 Jacob Bronowski
1939 James Wilkinson
1940 Hermann Bondi
1952 John Polkinghorne
1953 Crispin Nash-Williams
1959 Jayant Narlikar
1970 Derek…

By Simon Singh

Synopsis: ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.’

It was with these words, written in the 1630s, that Pierre de Fermat intrigued and infuriated the mathematics community. For over 350 years, proving Fermat’s Last Theorem was the most notorious unsolved mathematical problem, a puzzle whose basics most children could grasp but whose solution eluded the greatest minds in the world. In 1993, after years of secret toil, Englishman Andrew Wiles announced to an astounded audience that he had cracked Fermat’s Last Theorem. He had no idea of the nightmare that lay ahead.

In ‘Fermat’s Last Theorem’ Simon Singh has crafted a remarkable tale of intellectual endeavour spanning three centuries, and a moving testament to the obsession, sacrifice and extraordinary determination of Andrew Wiles: one man against all the odds.

First Published: 1997, Reissued: 2002| ISBN-13: 978-1841157917

Author’s…

► See the top performers in reading, mathematics and science  (on this page).

Chart A2·1 [ page 42] ranks countries, in descending order, according to the percentage of adults who have completed an upper secondary education (the most recent data in the 2013 report is from 2011).

PISA

• Singapore Math emphasizes the development of strong number sense, excellent mental-math skills, and a deep understanding of place value.
• The curriculum is based on a progression from concrete experience—using manipulatives—to a pictorial stage and finally to the abstract level or algorithm. This sequence gives students a solid understanding of basic mathematical concepts and relationships before they start working at the abstract level.
• Singapore Math includes a strong emphasis on model drawing, a visual approach to solving word problems that helps students organize information and solve problems in a step-by-step manner.
• Concepts are taught to mastery, then later revisited but not re-taught. It is said the U.S. curriculum is a mile wide and an inch deep, whereas Singapore’s math curriculum is said to be just the opposite.
• The Singapore approach focuses on developing students who are problem solvers.

I suppose every reader of this ‘ere blog will know Heron’s formula for the area $latex K$ of a triangle with sides $latex a,b,c$:

$latex K = \sqrt{s(s-a)(s-b)(s-c)}$

where $latex s$ is the “semi-perimeter”:

$latex \displaystyle{s=\frac{a+b+c}{2}.}$

The formula is not at all hard to prove: see the Wikipedia page for two elementary proofs.

However, I have only recently become aware of Brahmagupta’s formula for the area of a cyclic quadrilateral. A cyclic quadrilateral, if you didn’t know, is a (convex) quadrilateral all of whose points lie on a circle:

And if the edges have lengths $latex a,b,c,d$ as shown, then the formula states that the area is given by

$latex K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$

where as above $latex s$ is the semi-perimeter:

$latex \displaystyle{s=\frac{a+b+c+d}{2}.}$

This can be seen to be a generalization of Heron’s formula. Although the formula is named for Brahmagupta (598 – 670), who does indeed seem to…

Recently, in The Conversation, the Vice Chancellor of Monash University, wrote an article discussing MOOCs. He made some criticisms about the nature of assessment and grading that MOOCs offer. However, my attention was grabbed by two sentences:

The other major problem the MOOCs haven’t solved is assessment. They work very well for subjects like maths, which have objectively right and wrong answers, and can therefore be pretty easily marked by computers.

Now, here we have the Vice Chancellor of one of Australia’s leading universities – and indeed, one of the world’s leading universities (and incidently the University where I did both my Masters and my PhD) demonstrating an extraordinary lack of understanding about the fundamental nature of mathematics. He seems to think that mathematics is all about teaching students (in the fine words of John Power from Leeds University) about “finding ‘x'”. I suppose he thinks this…

WordPress.com supports LaTeX, a document markup language for the TeX typesetting system, which is used widely in academia as a way to format mathematical formulas and equations. LaTeX makes it easier for math and computer science bloggers and other academics in our community to publish their work and write about topics they care about.

If you’re a math genius — many of you are! — and you’ve blogged about equations you’ve worked on, you’ve probably used LaTeX before. If you’re just starting out (or simply curious to see how it all works), we’ve gathered a few examples of great math and computing blogs on WordPress.com that will inspire you.

In general, to display formulas and equations, you place LaTeX code in between $latex and $, like this:

$latex YOUR LATEX CODE HERE$

So for example, inserting this when you’re creating a post . . .

$latex i\hbar\frac{\partial}{\partial…

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.

2. By Quadratic Reciprocity,
$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…