Problem Archive

Consider the Gaussian integer $i-1$. A base $i-1$ representation of a Gaussian integer $a+bi$ is a finite sequence of digits $d_{n - 1}d_{n - 2}\cdots d_1 d_0$ such that:

• $a+bi = d_{n - 1}(i - 1)^{n - 1} + d_{n - 2}(i - 1)^{n - 2} + \cdots + d_1(i - 1) + d_0$
• Each $d_k$ is in $\{0,1\}$
• There are no leading zeroes, i.e. $d_{n-1} \ne 0$, unless $a+bi$ is itself $0$

Here are base $i-1$ representations of a few Gaussian integers:

$11+24i \to 111010110001101$
$24-11i \to 110010110011$
$8+0i \to 111000000$
$-5+0i \to 11001101$
$0+0i \to 0$

Remarkably, every Gaussian integer has a unique base $i-1$ representation!

Define $f(a + bi)$ as the number of $1$s in the unique base $i-1$ representation of $a + bi$. For example, $f(11+24i) = 9$ and $f(24-11i) = 7$.

Define $B(L)$ as the sum of $f(a + bi)$ for all integers $a, b$ such that $|a| \le L$ and $|b| \le L$. For example, $B(500) = 10795060$.

Find $B(10^{15}) \bmod 1\,000\,000\,007$.