## Archive for May 24th, 2012

### A look at a few Tripos questions VIII

May 24, 2012

Now for a question on modular arithmetic. As with countability, there is a very high chance of a question on this topic. [Added after the post was written: as usual I wrote down my thoughts about this question as I had them, and I didn't spot the best approach to part (ii) of the question until after I had come up with some less good approaches. So my recommendations evolve through the post, with some of the later ones superseding some of the earlier ones.]

6C. (i) Prove Wilson’s theorem: if $p$ is prime then $(p-1)!\equiv -1$ (mod $p$).

Deduce that if $p\equiv 1$ (mod 4) then

$\displaystyle \Bigl(\bigl(\frac{p-1}2\bigr)!\Bigr)^2\equiv -1$ (mod $p$).

(ii) Suppose that $p$ is a prime of the form $4k+3$. Show that if $x^4\equiv 1$ (mod $p$) then $x^2\equiv 1$ (mod $p$).

(iii) Deduce that if $p$ is an odd prime, then the congruence

$\displaystyle x^2\equiv -1$ (mod $p$)

has exactly two solutions (modulo $p$) if $p\equiv 1$ (mod 4), and none otherwise.
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### Have you signed the open-access petition?

May 24, 2012

Update 4th June 2012. The petition has now passed 25,000 signatures. It would still be great to push on and reach a significantly higher number by the June 19th deadline.

As you may know, there is a system in the US for setting up online petitions. Any petition that reaches 25,000 signatures in 30 days will be considered by White House staff. Recently, a petition was set up asking the Obama administration to require publications resulting from research paid for by the US taxpayer to be freely available. If such a requirement were to be put in place, it would be a huge boost to the campaign to make all academic research easily accessible.