Find common element of two arithmetic sequences that does not have differential?

Question

I have the following two sequences:

1. $a^n$
2. $b+cm$

I want to find the common element where the $n$ and $m$ meet in result.

Answer 1

I would suggest using modular algebra, and would demonstrate the thought process with the help of an example.

suppose $2^n=3+(5\times m)$
$\implies 2^n -3 = 5\times m$

we know that $2 \equiv 2 \pmod5$

$2^2\equiv -1 \pmod5$

$2^3\equiv -2 \pmod5$

$2^4\equiv 1 \pmod5$

$2^5\equiv 2 \pmod5$

and it cycles among these values forever after that

we can take out an important conclusion from the above statements, i.e

$$2^n \equiv -2 \pmod5 \ \forall \ n=3+4x : x \in \Bbb{Z}$$

Because we have:

$$3\equiv -2 \pmod5$$

And when we substract these these

$$2^n-3 \equiv 0 \pmod5 \ \forall \ n=3+4x : x \in \Bbb{Z}$$

Alternatively $$2^n -3 =5\times m \ \forall \ n=3+4x : x \in \Bbb{Z}$$

so $2^n$ can be expressed as $3+(5\times m) \iff n$ is of the form $3+4x$ where $x$ is an integer