| There is an apparent typo in the original problem. I assume it should read:
... the relation ... does *not* hold for any positive integer d <= n/(p-1).
The first few values for p=3 are:
a(0) = 0 (by definition)
a(1) = 0 (0 < d <= 1/2 is empty)
a(2) = 1 (if a(2) were 0 then a(2) = a(1) = a(0) would hold,
with d=1)
a(3) = 0 (a(3) = a(2) = a(1) doesn't hold)
a(4) = 0 (a(4) = a(3) = a(2) doesn't hold)
a(5) = 1 (if it were 0, then a(5) = a(4) = a(3) would hold;
a(5) = a(4) = a(3) doesn't hold;
a(5) = a(3) = a(1) doesn't hold)
a(6) = 1 (if it were 0, then a(6) = a(3) = a(0) would hold)
a(6) = a(5) = a(4) doesn't hold;
a(6) = a(4) = a(2) doesn't hold;
a(6) = a(3) = a(0) doesn't hold)
a(7) = 2 (if it were 0, then a(7) = a(4) = a(1) would hold;
if it were 1, then a(7) = a(6) = a(5) would hold)
a(7) = a(6) = a(5) doesn't hold;
a(7) = a(5) = a(3) doesn't hold;
a(7) = a(4) = a(1) doesn't hold)
a(8) = 2 (if it were 0, then a(8) = a(4) = a(0) would hold;
if it were 1, then a(8) = a(5) = a(2) would hold)
a(8) = a(7) = a(6) doesn't hold;
a(8) = a(6) = a(4) doesn't hold;
a(8) = a(5) = a(2) doesn't hold;
a(8) = a(4) = a(0) doesn't hold)
a(9) = 0 (a(9) = a(8) = a(7) doesn't hold;
a(9) = a(7) = a(5) doesn't hold;
a(9) = a(6) = a(3) doesn't hold; and
a(9) = a(5) = a(1) doesn't hold)
|
| The conjecture is false.
I'll counter that for p >=5, a[p^3-p^2+p-2] = 1, and that this is
the first a[n] for which the conjecture is false. This has been
verified for p <= 29.
Since a[r] = 0 for r < p-1, and a[p-1] = 1, we use the conjectured equation:
a[p*k+r] = [ (p*a[k] + r)/(p - 1) ] to see that:
a[p*(p-1)+0] = [ (p*a[p-1]+0)/(p-1) ] = [ p/(p-1) ] = 1
and
a[p*(p^2-p)+p-2] = [ (p*a[p^2-p]+p-2)/(p-1) ] = [ (2*p-2)/(p-1) ] = 2.
|