New Mersenne Prime

Monday 31 May 2004 at 2:32 pm | In Articles | Post Comment

It’s not often that one can report mathematics news but two weeks ago a new Mersenne prime was discovered. A Mersenne prime is a prime number of the form 2^p-1 ie one less than a power of two. It is easy to show that p itself must also be prime. The new prime number is 2^{24,036,583}-1 has 7,235,733 decimal digits and was found on a 2.4 GHz Pentium 4 computer running Windows XP.

You can help search for larger primes and possibly win $100,000 for discovering the first 10-million-digit prime. See the Great Internet Mersenne Prime Search (GIMPS)

LaTeX

Sunday 30 May 2004 at 1:18 pm | In Articles | Post Comment

\LaTeX is a very powerful language and allows one to type very nice mathematics in a way that would be difficult to do any other way. It is what provides the mathematics on this site. How else could one show

    \zeta(n)=\underbrace{\int_0^1 \cdots \int_0^1}_{n}\dfrac{\prod_{i=1}^n dx_i}{1-\prod_{i=1}^n x_i}

But \LaTeX will do more than that and also provide graphics. See the diagram in Monday 10 May or how about these:

    \setlength{\unitlength}{0.5cm}
\begin{picture}(5,2)
\put(5,0){\line(-1,0){5}}
\put(0,0){\line(1,1){2}}
\put(2,2){\line(3,-2){3}}
\put(-0.5,-0.1){\small A}
\put(1.9,2.1){\small B}
\put(5.1,-0.1){\small C}
\put(4,0.1){\tiny 30}
\qbezier(3.8,0)(3.9,0.55)(4.1,0.55)
\put(0.6,1){\footnotesize 2}
\put(3.6,1){\footnotesize 3}
\put(2.3,-0.6){\footnotesize 5}
\put(6,-0.2){\tiny \shortstack{t\r\i\a\n\g\l\e}}
\end{picture}

    go on – spot the deliberate error :-?

or

    \setlength{\unitlength}{1mm}
\begin{picture}(60,40)
\put(30,20){\vector(1,0){30}}
\put(30,20){\vector(4,1){20}}
\put(30,20){\vector(3,1){25}}
\put(30,20){\vector(2,1){30}}
\put(30,20){\vector(1,2){10}}
\thicklines
\put(30,20){\vector(-4,1){30}}
\put(30,20){\vector(-1,4){5}}
\thinlines
\put(30,20){\vector(-1,-1){5}}
\put(30,20){\vector(-1,-4){5}}
\end{picture}

See here for a good tutorial on \LaTeX graphics

Two Interesting Functions

Sunday 23 May 2004 at 10:33 am | In Articles | 2 Comments

At A level students only see reasonably nice functions, though occasionally they come across

    Graph of cosec

where the maximum is smaller than the minimum (which, incidentally, is why one should use the terms local maximum and local minimum instead).

But what they won’t have come across are functions like these:

    f(x)=\left\{\begin{array}{ll}1 & \mbox{ if } x \mbox{ is rational}\\0 & \mbox{ if } x \mbox{ is irrational}\end{array}\right.

    g(x)=\left\{\begin{array}{ll}1/q & \mbox{ if } x = p/q,\ q>0, \mbox{ in lowest terms}\\0 & \mbox{ if } x \mbox{ is irrational}\end{array}\right.

When trying to draw the graphs remember that

  • between every pair of irrational numbers there is a rational number
  • between every pair of rational numbers there is a irrational number

You can very roughly draw f using dots and there’s a nice picture of g at mathworld
Now think about integrating f and g – do they have areas under between them and the x-axis? The answer to ths question is important in the theory of integration.

These functions have other interesting properties and illustrate the relationship between rational and irrational numbers.

Nice Problem

Sunday 16 May 2004 at 7:28 pm | In Articles | Post Comment

There’s a very nice problem on The University of Warwick Mathematical Society’s site.
I won’t say exactly where because it has the solution there as well :-?

  1. If f is continuously differentiable on the open interval (a,b)
  2. \lim_{x \to a^{+}}f(x)=\infty
  3. \lim_{x \to b^{-}}f(x)=-\infty
  4. f^{\prime}(x)+f(x)^2 \geq 1

then prove that b-a \geq \pi

Stuck!. Then click on read more for a hint
Continue reading Nice Problem…

Going dotty

Monday 10 May 2004 at 6:59 pm | In Articles | Post Comment

A well-known problem is to be given 9 dots

    \setlength{\unitlength}{0.5cm}
\begin{picture}(7,7)
\multiput(0,0)(1,0){3}{\circle*{0.15}}
\multiput(0,1)(1,0){3}{\circle*{0.15}}
\multiput(0,2)(1,0){3}{\circle*{0.15}}
\thinlines
\end{picture}

You have to connect the dots by using 4 lines, without taking your pen off the paper and only going through each dot once. At first sight this looks impossible until you are shown you can go outside the square formed by the dots as in

    \setlength{\unitlength}{0.5cm}
\begin{picture}(7,7)
\multiput(0,0)(1,0){3}{\circle*{0.15}}
\multiput(0,1)(1,0){3}{\circle*{0.15}}
\multiput(0,2)(1,0){3}{\circle*{0.15}}
\thinlines
\put(1,2){\line(-2,0){2}}
\put(-1,2){\line(1,-1){3}}
\put(2,-1){\line(0,1){3}}
\put(2,2){\line(-1,-1){2}}
\end{picture}

But what about 16 dots arranged in a square, or 12 dots arranged in 3 rows of 4? How many lines are needed for a 10 x 10 grid? What if arcs of circles are used instead of lines?

These and other fascinating questions about dots can be found at the web page simply called dots

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