# Problem 558: Irrational base
Let r be the real root of the equation x3 = x2 + 1. Every positive
integer can be written as the sum of distinct increasing powers of r. If
we require the number of terms to be finite and the difference between
any two exponents to be three or more, then the representation is
unique. For example, 3 = r -10 + r -5 + r -1 + r 2 and
10 = r -10 + r -7 + r 6. Interestingly, the relation holds for the
complex roots of the equation. Let w(n) be the number of terms in this
unique representation of n. Thus w(3) = 4 and w(10) = 3. More formally,
for all positive integers n, we have:n = \$\\displaystyle
\\sum\_{k=-\\infty}\^{\\infty}\$ bk rk under the conditions that:bk is 0
or 1 for all k;bk + bk+1 + bk+2 ≤ 1 for all k;w(n) = \$\\displaystyle
\\sum\_{k=-\\infty}\^{\\infty}\$ bk is finite. Let S(m) =
\$\\displaystyle \\sum\_{j=1}\^{m}\$ w(j2). You are given S(10) = 61 and
S(1000) = 19403. Find S(5 000 000).