Factors of Sums of Powers of 100,000/General Result

Theorem
All integers $n$ of the form:
 * $n = \displaystyle \sum_{k \mathop = 0}^m 10^{r k}$ for $m \in \Z_{> 0}$

are composite for $r \ge 2$.

The only exceptions are $r = 2^k, m = 1$ for some $k \in \N$,

and $r = m + 1 =$ some odd prime, where $n$ could be prime.

Case $1$: $m + 1$ is composite
Suppose $m + 1$ is composite. Then:


 * $\exists p, q > 1: m + 1 = p q$

By Division Theorem, for each $k$ with $0 \le z \le m$:


 * $\exists i, j \in \N: 0 \le i \le q - 1, \, 0 \le j \le p - 1: k = i + q j$

Thus:

Both sums are greater than $1$, so $n$ is composite.

Case $2$: $m + 1$ is an odd prime and $r \ne m + 1$
Notice that $\displaystyle \sum_{k \mathop = 0}^m 10^{r k} \times R_r = R_{r \paren {m + 1} }$, where $R_i$ is the $i$th repunit.

Suppose $m + 1$ and $r$ are coprime.

By Condition for Repunits to be Coprime, $R_{m + 1}$ and $R_r$ are coprime.

By Euclid's Lemma, $R_{m + 1} \divides \dfrac {R_{r \paren {m + 1} } } {R_r} = n$.

Suppose $r^2 \divides m + 1$.

By Divisors of Repunit with Composite Index, $R_r \divides R_{r^2}$, and $R_{r^2} \divides R_{r \paren {m + 1} }$.

So we have $\dfrac {R_{r^2}} {R_r} \divides \dfrac {R_{r \paren {m + 1} } } {R_r} = n$.

Case $3$: $m = 1$, $r$ is not a power of $2$
We write $n = 1 + 10^r$.

Since $r$ is not a power of $2$, $r$ has an odd factor greater than $1$.

Write $r = x \paren {2 y + 1}$.

Then:

Since $1 + \paren {-1}^{2 y + 1} = 0$,

by Polynomial Factor Theorem, $1 + z^{2 y + 1} = \paren {1 + z} \map Q z$ for some polynomial $Q$.

Therefore $1 + 10^x$ is a factor of $1 + \paren {10^x}^{2 y + 1} = n$.

Thus all cases are covered.