Triskaidekaphilia (Part II)

6. 13 is a prime such that for any digits a,b > 0 and any integers n,m > 1 one can build an integer divisible by it having exactly n digits equal to a, exactly m digits equal to b and exactly 2 digits equal to 0. Moreover this m+n+2 digit long number is non-trivial, in that none of the two 0 digits is located at the end of the number.

Indeed this is true since:
\(13 \mid 10\underbrace{1 \ldots 1}_{k \; times \; 1}01, \forall k \in \mathbf{N}\)

One other prime with this property is 7. This can easily e extended to any number of digits.

7. 13 is a prime with the property that:

\( \forall a,b \in \mathbf{N} \; \left\{ \begin{array}{l} p \mid \overline{ab} \\ p \mid \overline{bc} \end{array} \right. \rightarrow p \mid \overline{ca} \)

This is equivalent to the fact that the following equation system in a, b and c:

\( \left\{ \begin{array}{ccccccc} 10 \cdot a & + & 1 \cdot b & + & 0 \cdot c & = & 0 \\ 0 \cdot a & + & 10 \cdot b & + & 1 \cdot c & = & 0 \\ 1 \cdot a & + & 0 \cdot b & + & 10 \cdot c & = & 0 \end{array} \right. \)

has non-trivial solutions modulo p, hence for its discriminant we have that:

\( \left| \begin{array}{ccc} 10 & 1 & 0 \\ 0 & 10 & 1 \\ 1 & 0 & 10 \end{array}\right| = 1001 \equiv 0 \mod{p} \)

Therefore the primes with this property are 7, 11 and 13.

8. 13 is a prime with the property that:

\(\forall a,b \in \mathbf{N} \; \left\{ \begin{array}{l} p \mid \overline{ab} \\ p \mid \overline{bc} \end{array} \right. \rightarrow p \mid a + b + c\)

9. 13 is a prime with the property that:

\(\forall a,b \in \mathbf{N} \; \left\{ \begin{array}{l} p \mid \overline{ab} \\ p \mid \overline{bc} \end{array} \right. \rightarrow p \mid a^2 + b^2 + c^2\)