## Archive for April, 2007

### Intuitionistic logic

April 27, 2007

These days I’ve read (part of) a nice introductory book in french by Pierre Ageron on intuitionistic logic.

I haven’t read the later chapters on category theory, about which I’ll only remember the following sentence:

[…] one distinguishes small categories, those for which the classes $C$ of objects and $F$ of arrows are sets, from large categories, which are all the others. Beyond merely the language of categories, common to both small and large categories, it’s in reality the dialectic articulation between small and large categories which does constitute the heart of category theory.

I was more interested by the beginning of the book, which I’ll summarize for myself as follows.

### A little bit of arXiv humor

April 26, 2007

arXiv:quant-ph/0512170 [ps, pdf, other] :
Title: Error correcting codes for adiabatic quantum computation
Authors: Stephen P. Jordan, Edward Farhi, Peter W. Shor
Journal-ref: Phys. Rev. A 74, 052322 (2006)

### Some azur (3)

April 20, 2007

Miremont is a small village near Toulouse. Its surroudings look really gorgeous according to this picture

which comes from FreePhotoBank.org and is under CC licence.

### The Giuga-Agoh conjecture and additive combinatorics

April 18, 2007

In 1950, Giuseppe Giuga conjectured that $p$ is prime iff $1+1^{p-1}+2^{p-1}+\dots + (p-1)^{p-1} \equiv -1\ (\mod p)$. It was later shown by Takashi Agoh that it is equivalent to the conjecture he had made independently $p$ is prime iff $pB_{p-1}\equiv -1\ (\mod p)$ where the $B_k$ are Bernouilli numbers, thus becoming the Giuga-Agoh conjecture.

Borwein et al. have shown that the conjecture is true for numbers with at least up to 13800 digits (yes, that means the conjecture is true for all primes less than something like billions of billions of billions of billions…). More recently Vicentiu Tipu has shown (using methods similar to those used to deal with Carmichael numbers) that the number $G(X)$ of possible counterexamples to the conjecture (i.e. composite numbers satisfying the congruence) at least doesn’t grow too much at infinity: $G(X) \ll \sqrt{X}\log X$, for large $X$.

Now, stated with the congruence above, the conjecture might seem to the reader to be yet another of those probably-true-but-uninteresting statements that one encounters in elementary number theory. Yet, I’d like to think it’s not the case.