Category Archives: Number Theory

142857*2=285714 142857*3=428571 …

Let , then (And 7a=999999. ) This seems to be an amazing property of . But then, why ? Advertisements

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A dynamical proof of Fermat’s little theorem

If you have studied number theory (even just a little bit), you should know the Fermat’s little theorem: for any positive , and for any prime number , we have . Here I will give a proof by using a … Continue reading

Posted in Dynamical system, Number Theory | 3 Comments

Poincaré recurrence theorem and its friends

The Poincaré recurrence theorem states that a dynamical system (under suitable conditions) will eventually return to a condition that is very close to the original condition. Slightly more formally, for a compact set in , if is a volume preserving … Continue reading

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Uniform distribution of polynomials mod 1

Last week Lam Wai Kit posted an excellent introductory article of ergodic theory and its applications to number theory. As stated in the note, one consequence of Birhoff recurrence theorem is that for any non-constant polynomial , the fractional part … Continue reading

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A dynamical proof of some Diophantine approximations

Diophantine approximations is a field in number theory which asks how good a real number can be approximated by some rational number. Quite a number of classical results are proved by using combinatorics, which is not friendly to people who … Continue reading

Posted in Dynamical system, Number Theory | 2 Comments

Arithmetic progression in some subsets of $\mathbb{Z}_N$ (Part 2)

Last time I have mentioned the idea of the proof (of theorem 2). Now I continue and give a full detail here.

Posted in Fourier analysis, Number Theory | 1 Comment

Arithmetic progression in some subsets of $\mathbb{Z}_N$ (Part 1)

This post is based mainly on the paper “Arithmetic progression in sumsets” by Ben Green, which talks about the length of arithmetic progressions of some special kind of subsets of , where is a sufficiently big odd prime. I will … Continue reading

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