Dr Who’s Happy Primes

Saturday 19 May 2007 at 8:34 pm | In Articles | 1 Comment

It’s so nice to see mathematics playing an important part in popular television. In tonight’s Dr Who the spaceship crew had to find the next number in the sequence 313, 331, 367, …. Dr Who recognises this as a sequence of happy primes with the next one being 379. It’s all explained at that Wikipedia article and a longer version of the sequence can be found at the wonderful The On-Line Encyclopedia of Integer Sequences.

It’s interesting that the Dr Who reference was put in the Wikipedia article half-an-hour before the programme was aired so probably an inside job. I was delighted that in the programme Dr Who asked if mathematics was so dumbed down that recreational maths wasn’t studied any more. As Russell T. Davies, the head writer, said in tonight’s Dr Who Confidential, the programme reflects current concerns, so this problem has clearly reached a wider audience than I imagined was the case.

Eigenvalues without determinants

Friday 18 May 2007 at 3:14 pm | In Articles | 9 Comments

Most (all?) undergraduate courses use determinants to introduce eigenvalues and eigenvectors. So the eigenvalues of a matrix A (or linear transformation t) are the solutions of \det(A-\lambda I)=0. However, Sheldon Axler published a paper in 1994 called Down with Determinants! where he maintains that determinants should not be used so early in linear algebra courses. He gives a very nice proof of the existence of eigenvalues in finite-dimensional vector spaces (over \mathbb{C}) which I would like to reproduce here.

Every linear transformation of a finite-dimensional complex vector space has an eigenvalue.

Here is his proof:

Let V be a non-trivial finite-dimensional complex vector space and t a linear transformation t\colon V \to V. Let v be a fixed non-zero vector in V and suppose that \dim V =n. Then the n+1 vectors v,t(v),t^2 (v),\dots,t^n (v) are linearly dependent. Hence there exists complex numbers \alpha_0,\alpha_1,\dots,\alpha_n not all 0 such that

\alpha_0 v + \alpha_1 t(v) + \dots + \alpha_n t^n (v)=0.

and hence

(\alpha_01+ \alpha_1 t + \dots + \alpha_n t^n)(v)=0.

Now, since \mathbb{C} is algebraically closed,  the polynomial \alpha_0 + \alpha_1 z+ \dots + \alpha_n z^n will factorise so we get

\alpha_0 + \alpha_1 z+ \dots + \alpha_n z^n=c(z-\lambda_1)(z-\lambda_2)\dots(z-\lambda_m).

where c, \lambda_1,\dots,\lambda_m are complex numbers with c \neq 0. It follows that

c(t-\lambda_1 1)(t-\lambda_2 1)\dots(t-\lambda _m 1)(v)=0

which means that, since this is composition of functions, and v \ne 0, then
either
(t-\lambda _m 1)(v)=0 so  (t-\lambda_m 1)(v)=v so that t(v)=\lambda_m v and v is an eigenvector
or
(t-\lambda _{m-1} 1)(t-\lambda _m 1)(v)=0 so (t-\lambda _m 1)(v) is an eigenvector
or

or
(t-\lambda_2 1)\dots(t-\lambda _m 1)(v) is an eigenvector,
and hence t has an eigenvalue.   \blacksquare

Discussion on this approach of not using determinants can be found at NeverEndingBooks and The n-category Café.

Problems, Problems

Saturday 5 May 2007 at 2:22 pm | In Articles | 3 Comments

The Art of Problem Solving (AopS) site encourages, teaches and promotes mathematics competitions from small local ones right up to the International Mathematical Olympiad (IMO). As it says on its front page:

Is math class too easy for you? Looking for a greater challenge?
You’ve come to the right place.

It has an excellent introduction to \LaTeX site and a forum to discuss problems. The forum has an RSS feed where students pose new problems every day. So if you’re “looking for a greater challenge” subscribe to this feed. Here is a random sample of some of the problems posed recently, which range from very easy to extremely difficult. Click on the problem number to go to the discussion on it.

    1. We define addition in a different way to usual; an addition statement is true only if the letters in the addends is a rearrangement of the letters in the sum. For example,
    10 + 6 = 16?
    TEN + SIX = TENSIX = SIXTEN, but to be 16 it would need another E.
    Find a “true” addition a + b = c + d.

    2. Prove that 1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots<2.8.

    3. Let a,b,c be nonzero real numbers. Find all ordered pairs (a,b,c) such that \displaystyle\frac{2(a-b-c)}{a^{2}}=\frac{4b-a-2c}{b^{2}}=\frac{4c-a-2b}{c^{2}}.

    4. f is a continuous complex-valued function satisfying:
    i) |f(z)| = |z|
    ii) |f(z)-z| = |z|
    Find f(f(f(2007)))

    5. If a and b are relatively coprime, find all possible values of \gcd(a+b,a^{2}+b^{2}).

    6. Let A,B,C be three angles of  \triangle ABC. Prove that (1-\cos A)(1-\cos B)(1-\cos C)\ge\cos A\cos B\cos C.

    7. For each function f which is defined for all real numbers and satisfies f(xy)=xf(y)+yf(x) and f(x+y)=f(x^{1993})+f(y^{1993}) determine the value of f(\sqrt{5753}).

    8. Let a,b,c be positive reals such that a+b+c=1. Prove that 7(ab+bc+ca) \le 9abc+2.

    9. For 0 \leq d \leq 9, we define the numbers S_{d}=1+d+d^{2}+\cdots+d^{2006}. Find the last digit of the number S_{0}+S_{1}+\cdots+S_{9}.

    10. The product of several distinct positive integers is divisible by 2006^2. Determine the minimum value the sum of such numbers can take.

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