**Primes:**
$100..00300..00900..009$ is prime when the number of zeros between consequtive digits are aqual to $0, 2, 350, 822 \leq 2000$

**Dividibility:**
$100..00300..00900..009 \equiv 0 \pmod{1399}$ when the number of zeros between consequtive digits are equal to $0, 17, 699, 716, {\bf 1398}, 1415, 2097, 2114, 2796, 2813, 3495, 3512, 4194, 4211, 4893, 4910, 5592, 5609, 6291, 6308, 6990, 7007, 7689, 7706, 8388, 8405, 9087, 9104, 9786, 9803 \leq 10000$

$S_{1400}(100..00300..00900..009) \pmod{1399} = 0$, where the number of zeros between consequtive digits is equal to $1398$.

**Diopantine equations:**
The number of solutions to the equation $\frac{4}{765}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is equal to $1399$.