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Given a prime $p$ and an integer $0 \leq k \leq p-1$, a famous congruence on binomial coefficients states: $$\binom{p-1}{k} \equiv (-1)^k \pmod{p}$$ It is generally taught as a consequence of Pascal’s rule (1653-1665) and Wilson’s theorem (1770-1771).

We need someone expert in History of Mathematics (especially about the subject 11B65) to answer the following three questions:

  1. Who is credited for its discovery (i.e., who left the first clear statement about it)?
  2. Who established it (i.e., who supplied its first known proof)?
  3. Was it originally proved as above or via alternative formulations like, e.g., $1+(-1)^{k+1} \binom{p-1}{k} \equiv 0 \pmod{p}$?

Thank you for your help.

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    $\begingroup$ If you don't get an answer here you might try hsm.stackexchange.com $\endgroup$
    – Michael Lugo
    Commented Feb 8 at 15:25
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    $\begingroup$ A more known result is that $p$ divides $\binom{p}{k}$ whenever $1 \le k \le p-1$. The congruence you give is a simple consequence (by recursion on $k$), thanks to the recursion relation satisfied by binomial coefficients. So Pascal probably knew it. $\endgroup$
    – Christophe Leuridan
    Commented Feb 8 at 15:25
  • $\begingroup$ Thank you @ChristopheLeuridan. It is so simple that its historical origin is very hard to find. $\endgroup$
    – Monk
    Commented Feb 8 at 15:35
  • $\begingroup$ Thank you for your advice @MichaelLugo. We are confident that something helpful and accurate will be written here within 24h. $\endgroup$
    – Monk
    Commented Feb 8 at 15:47
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    $\begingroup$ @ChristopheLeuridan There is a more direct proof: $(p-1)\dotsb (p-k)\equiv (-1)\dotsb(-k)=(-1)^k k!\pmod{p}$. $\endgroup$
    – GH from MO
    Commented Feb 9 at 2:13

1 Answer 1

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Dickson in his "History of the theory of numbers. Vol. I: Divisibility and primality." (see page 64) attributes this statement to Genty (Histoire et mem. de I'acad. roy. sc. insc. de Toulouse, 3, 1788 (read Dec. 4, 1783), p. 91).

Before this he descibes the history of Wilson's theorem.

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  • $\begingroup$ Thank you @Alexey Ustinov, it is a great improvement. It seems that between Lagrange's proof of Wilson's theorem and Genty's statement there was more than a decade without anyone noticing such property... Difficult to believe. Should you have further informations about the period 1772-1783, even with different names from Lagrange and Genty, please enrich the discussion with another answer of yours. $\endgroup$
    – Monk
    Commented Feb 8 at 16:15

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