Quintics x5-5x-k, the Golden Section, and Square Lucas Numbers

Here we consider a special case of the problem of characterizing the real numbers that, once multiplied by the integer a, have the same fractional parts as those of their n th powers (n ≥ 2 an integer). These numbers, which will be referred to as numbers possessing the property p(n,a), are clearly given (cf (1.1) of [3]) by the real roots of the equations $$x^n - ax = k$$ , where the integer k can be assumed to be nonnegative without loss of generality. It is quite obvious that all the integers possess p(n,a) for all n and a: they emerge from (1.1) when k = m n - am (m an integer). The closed-form expressions for the above mentioned numbers can be readily found for 2 ≤ n ≤ 4 by using (1.1) and the well-known formulas for the solution of second-, third- and fourth-degree equations. Finding them for n ≥ 5 is a much more difficult task. The expressions for the numbers possessing p (n, 1) were worked out in [3] (see also [4]) for n ≤ 5. The special case n = 5 shows an interesting connection with Fibonacci numbers.