Novel power series

Question

Has anyone collected such novelties:
$1+5+25+125=156=12\cdot13$;
$1+2+4+8=15=3\cdot5$;
$1+2+4+8+16+32=63=7\cdot9$;
$1+18+324=343=7^3$;
$1+3+9+27+81=121=11^2$?

Answer 1

These observations can be summarized with the (well-known) formula $$ 1 + x + \cdots + x^n = \frac{x^{n+1} - 1}{x - 1}. $$ For instance, with $x=3$ and $n=4$ we obtain $$ 1 + 3^1 + \cdots + 3^4 = \frac{3^5 - 1}{3 - 1} = \frac{242}{2} = 121 = 11^2 $$ which confirms your observation.

The fact that such numbers have predictable possibilities for factorization is relevant to what we know about Mersenne primes. Mersenne primes are precisely the prime numbers that we can obtain when we take $x = 2$ in the above; notably such a number can only be prime if $n+1$ is prime, but the converse fails to hold.

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You may find it interesting that whenever $n+1$ is composite (non-prime), the quantity $\frac{x^{n+1} - 1}{x-1}$ has a corresponding factorization. In partiular, we find that if $n+1 = pq$ for integers $p,q > 1$, then $$ 1 + x + \cdots + x^n = \frac{x^{n+1} - 1}{x-1} = \frac{(x^{p})^q - 1}{x-1} = \frac {x^p - 1}{x-1} \cdot \frac{(x^p)^q - 1}{x^p - 1} = \\ \frac {x^p - 1}{x-1}\cdot [1 + x^p + \cdots + (x^p)^{q-1}] = \\ [1 + x + \cdots + x^{p-1}] \cdot [1 + x^p + \cdots + (x^p)^{q-1}] $$