here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

64 Chapter 2. On a conjecture about addition modulo 2 k − 1where⎧⎨ A j + Bj+1j+1if j > 0C j = −⎩13 6if j = 01 if j = −1The exact values of I and J are not important, only their cardinalities are; so defining I ={m + 1, . . . , m + i} and J = {n + 1, . . . , n + j}, we get( ) n−m ((i = ln − m1,...,i n) (−1)n+1 i 1 , . . . , i n ia d,n⎛⎝ ∑ k≥1∑i=02 k(h − k)!) d−n∑j=0( d − nj).∑k j≥0,j∈I∪Jk j≥i j−1,1≤j≤mS!l![ ] ⎞h − k ⎠ ∏ A kj∏ A kj − 3 kj=0S k j ! k j !j∈J j∈Im∏j=1C kj−i j|k j − i j |!We finally make the change of summation variables k j = k j − i j + 1 to obtain the desiredexpression:( ) n−m(i = l ∑( n − m1,...,i n) (−1)n+1 i 1 , . . . , i n ii=0⎛a d,n⎝ ∑ k≥1( )= (−1) n+1 li 1 , . . . , i n) d−n∑j=0( d − nj)∑k j≥0,j∈I∪J,1≤j≤m2 k [ ] ⎞ h − k ⎠ ∏ A kj∏ A kj − 3 kj=0(h − k)! l + S − m k j ! k j !j∈J j∈Ib d,nl,m .2.5.6 An additional relationIn this subsection we prove the following experimental fact.Proposition 2.5.3. For n ≥ 2 and 0 < j ≤ i,(i,j,...) = i + 1 a d,nj(i+1,j−1,...) ;a d,ni.e. the value of b d,nl,mdoes not depend on m.Proof. From Proposition 2.5.2,( )(i = l1,...,i n) (−1)n+1 i 1 , . . . , i na d,nb d,nl,m ,(l + S − m)!l!m∏j=1.C kj−1|k j − 1|!where b d,nl,monly depends on d, n, l and m. Therefore if j > 1, this value does not vary and thetheorem is a simple corollary of Proposition 2.5.2.If there is some degree equal to zero in (i, j, . . .), i.e. if n > m, then we can use the result ofCorollary 2.5.18:a d,n(i,j,...,0) + ad,n−1 (i,j,...) = 3ad−1,n−1 (i,j,...);hence we can restrict ourselves to the study of tuples where n = m.