If we have a modular equation say $5^{x+1}\mod2^{n+1}$, where $5^x \equiv1\mod2^n$, we also know that $5\equiv1\mod2$. I know from playing around with an online mod calculator that $5^{x+1}\equiv1\mod2^{n+1}$ aswell.
My question is:
Do we have some type of rule for this maybe something like $5^{x+1}\mod2^{n+1}\equiv5^x\mod2^n \times 5\mod2$ ?
On
here's a few modular arithmetic rules: $$y\equiv b \bmod m\implies y=mx+b$$ and $$y_1\equiv b_1\mod m\land y_2\equiv b_2\bmod m\implies y_1y_2\equiv b_1b_2\bmod m$$ expanding the polynomial form with the two y values equal( forcing the b's to be). we get: $$y_1^2=m^2x^2+2mxb_1+b_1^2$$ noting that doubling m raises the exponent by 1 on the base 2 side of things, we get $$5^2\equiv 1\bmod 2^2$$ FOIL is still obeyed. Just keep FOILing and checking.
There can't be an equation as take $n=3$ $5^2\cong5^4\cong1$ mod8 yet $5^3\cong13$ mod 16 while $5^5\cong1$. This problem is essentially the fact that 5 always generates the largest non trivial subgroup of $\mathbb{Z}/2^n\mathbb{Z}^\times\cong C_2\times C_{n-2}$ so we can always take $x=2^{n-2}=\varphi(2^{n-1})$ and $y=2^{n-1}=\varphi(2^n)=2\varphi(2^{n-1})$ and thus as 5 generates we see that $5^x\ncong5^y\cong1$mod $2^n$. However, should you put a restriction on $x$ I'm sure you should make an equation.