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Abstract: In the 1870s, Lucas obtained a beautiful formula for binomial coefficients modulo p. Namely, Binomial[n, m] is congruent modulo p to the product of the binomial coefficients whose arguments are the base-p digits of n and m, taken pairwise. Variations and generalizations of this formula have been actively investigated since. In particular, for which n and m does Lucas' congruence hold not just modulo p but modulo p²? The answer is related to some hidden rotational symmetry in Pascal's triangle. A similar result holds for the Apéry numbers, which Gessel showed satisfy a Lucas congruence in 1982

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password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420