"quasi" <email@example.com> a écrit dans le message de news: firstname.lastname@example.org... > --- (9) lemma > > If n is an odd positive integer such that q = (n-1)/2 is an odd > prime, then A(n,k) = 0 mod q for all even integers k with > 2 <= k <= q-1. > > Proof outline: > > The verification for k = 2 is immediate. Proceeding by induction > on k, assume k is even with 2 < k <= q-1 and simplify the > difference A(n,k) - A(n,k-2). It's not hard to show (not hard > but a little tedious) that, assuming 2 < k <= q-1, the difference > is congruent to 0 mod q, hence the truth of the lemma follows > from its truth for k=2. > > quasi
(9) Note that if q is prime et k is an integer with 2 <= k <= q-1, it is easy to prove that Binomial(2q+1,k) an Binomial(q,k) are divisible by q without induction. For this , the hypothesis 2q+1 prime is not necessary.