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408 Notes on Explorations

(i.e. i!” - fk(a)t + a&l(a) is a multiple of t2 - t + a) where fl(a) = 1,

fi(a) = 1 and fk+l(U) = fk(a) - afk-l(a) for k 2 2. Then t” + t + b is

divisible by t2 - t + a if and only if fn(a) + 1 = 0 and b = uf,-l(u). Hence

we need to examine the values of n and a for which f,,(a) = -1. When

a = 1, the situation is straightforward and fn(l) = -1 _ n = 6k + 4 or

6k + 5 for some 6. For n = 6k + 4, we have that

t2 - t + 11 t6k+4 + t = t(t3 + l)(Pk - * *. + 1).

For n = 6k + 5, we have that

t2 - t + 11 t6k+5 + t - 1.

This is easily checked since

t5 + t - 1 = (t2 - t + l)(P + t2 - 1)

and

t6”+5 + t - 1 = f+‘)+5(t6 _ 1) + tW--‘I+5 + t _ 1

for k 2 1.

For higher values of a, the problem becomes more interesting. fn(2) = -1

at least for n = 3, 5, 13, and we obtain

t3 + t + 2 = (t2 - t + 2)(t + 1)

t5 + t - 6 = (t2 -2 + 2)(t3 +t2 -t - 3)

t13 + t + 90 = (t2 - t + 2)(P + t1° - tg - 3ts - t7 + 5t6

+ 7t5 - 3t4 - 17t3 -llt2+23t+45).

The question of the existence of other solutions to the equation f”(2) = -1
is apparently quite difficult. It is known that a second order recurrence like
fn(a) for a # 1 visits a given number at most finitely often. See the following
papers :

R. Alter & K. Kubota, Multiplicities of second order linear recurrences
Trans. Amer. Math. Sot. 178 (1973), 271-284.
K. Kubota, On a conjecture of Morgan Ward, I
Acta Arithmetica 33 (1977), 11-48.
R. Loxton, Linear recurrences of order two
J. Austml. Math. Sot. 7 (1967), 108-114.
M. Mignotte, A note on recursive sequences
J. Austml. Math. Sot. 20 (A) (1975), 242-244.
For the special case n = 5, see Stanley Rabinowitz, The factorization of
x5 f x + n, Math. Mug. 61 (1988), 191-193.