First I tried some identities but could not find a way to make it work:

x^2-x-1 = 0

x - 1/x = 1

x^2 = x+1

ax^17+bx^16+1 = (ax+b)*(x^16) + 1

 = (ax+b)*((x+1)^8) + 1

I started doing long division, and noted a pattern where the a,b terms add like Fibonacci.  I thought I had the pattern figured out, and eventually found x^2 - x - 1 dividing into (987a + 610b)x^2 + (610a + 377b)x + 1

which might only be true if 

987a + 610b = -1 and 

610a + 377b = -1

377a + 233b = 0

So, maybe:

a,b could be 233, -377:  a-b = 610

a,b could be -233, +377:  a-b = -610

But when I plugged (233, -377) into Wolfram Alpha, I got a remainder of (2x+2)

...  which looks close, so I thought I may have miscounted the Fibonacci numbers.

So I tried other Fibonacci numbers.

Plugging in (377, -610) shows a remainder of -x

Plugging in (610, -987) shows a remainder of x+1

Plugging in (987, -1597) shows a remainder of 0

However trying (987, -1597) in Wolfram Alpha does not show a remainder of 0; instead complex solutions.  So I'll reject this one

Apparently:

(a,b) = (987, -1597)  and (a-b) = 2584

I must have made a math error early on.