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Section $2$ of AKS-Algorithm paper discusses the Idea of the Algorithm.

The paper uses the notation in equation $(2)$: $$(X+a)^n=X^n+a(\text{mod}\enspace X^r-1,n)$$

Does this notation mean $$[((X+a)^n-(X^n+a)\enspace\text{mod}\enspace n)\text{mod}\enspace X^r-1]$$

Also, how does it immediately follows that all primes satisfy the above equation? How can I prove "Mathematically" the two lines comments that succeed equation $(2)$ in the above paper?

Kumar
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  • Are you aware that this paper is not fully reliable when you see that many errors have been found in a previous version... – Jean Marie Jan 12 '20 at 16:06
  • @JeanMarie No. I didn't know. Can you please provide me with latest version and also the reliable links for the algorithm? Thanks. – Kumar Jan 12 '20 at 17:45
  • My previous remark was maybe not well grounded... forget it. As I am not specialized in this area, I have no alternative to propose you. – Jean Marie Jan 12 '20 at 17:58
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    The first dupe explains the notation. Primes satisfy it because the binomial coefficient $,(p\choose k)$ is divisible by $p$ for $,1\le k\le p-1,$ (see 2nd dupe) and because $,a^p\equiv a\pmod{p},$ by Fermat's little theorem. – Bill Dubuque Jan 12 '20 at 19:17

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