(In reply to
Long journey with an error; but got the answer by Larry)
Title of my post is wrong - should read
"Same answer but with a shortcut"
Also, I note a small typo in Larry's post.
I assigned x to be phi, the Golden Ratio and used:
x^n = F(n) x + F(n-1). So,
a x^17 + b x^16 + 1
= x (a x^16 + b x^15) + 1
= x [ a (F16 x + F15) + b (F15 x + F14) ] + 1
= (987 a +610 b) x^2 + (610 a + 377 b) x + 1 (in agreement with Larry)
The constant term 1 requires a single solution with a quotient of -1:
-1 (x^2 -x -1)
= - x^2 + x + 1
= (987 a +610 b) x^2 + (610 a + 377 b) x + 1
so,
987 a + 610 b =-1
610 a + 377 b =+1 (typo in Larry's post, he wrote -1 but used +1)
solving, again, gives:
a = 987, b = -1579
a - b = 2584
Edited on August 13, 2024, 1:29 pm
different answer....
