A result about Fermat's numbers. Is my proof correct? Is that result useful ?Can we generalize that result?

Question

Let $n$ be an integer and $F_n:=2^{2^n}+1$. $$n=2,3,4:F_n\equiv17\pmod{30}$$

$\mathbf{Result:}\;n>1:F_n\equiv17\pmod{30}$

$\mathbf{Proof:}$ Suppose $F_n - 1\equiv16\pmod{30}$. Then $2^{2^{n+1}}=({2^{2^n}})^2\equiv16^2\pmod{30}\equiv16\pmod{30}$ and $F_{n+1}\equiv17\pmod{30}$.

I found that result using primoradic (see stub OEIS: https://oeis.org/wiki/Primorial_numeral_system).

That's the way I found that result, which I did't know before and I wonder if there are other results with $\;210,\;2310,\;30030,\;\ldots$ (primorials)

P.S. : In primoradic, using Charles-Ange Laisant's notations for factoradic (with $A=10, B=11, C=12, D=13,\ldots$) $$17=(000000.221)$$ $$257=(000011.221)$$ $$65.537=(2.240.221)$$ $$F_5=(J.5F1.721.221)$$ Perhaps, someone could give $F_6$, $F_7$ ...in primoradic, just for fun. Fill in the holes :$$F_6=(........0.221)$$ $$F_7=(......1:1:...)$$ $$F_8=(......4:0:...)$$ $$F_9=(......2:1:...)$$ $$F_{10}=(....0:0:...)$$ $$F_{11}=(......1:...)$$ $$F_{12}=(......0:...)$$ $$F_{13}=(......1:...)$$ $$2^{16384}+1=(......:0:...)$$ I have verified each of these results with my spreadsheet. Maybe $F_n\equiv17\pmod{210}$ if n is even and $47$ if n is odd? A result appears clearly with $2310$ too. We need a proof. Perhaps here :I'm trying to generalize some simple results about $ 2 ^ n $. It's useful to write them in primoradic (see stub OEIS)., will be useful.

Answer 1

Primorials play no role. Rather, fixed points of (quadratic) polynomial iterations are key.

Notice $\ \color{#0a0}{F_{n+1} = (F_n-1)^2+1} = F_n^2 -2F_n + 2\ $ hence $\!\bmod 30\!:\ F_{n+1}\equiv F_n\iff 0 \equiv F_n^2-3F_n + 2 \equiv (F_n-1)(F_n-2).\,$ Using CRT to combine the roots $\,F_n\equiv 1,2\,$ mod $\,2,3,5\,$ (as here) yields $\,2^3\,$ roots $\,1,2,7,11,16,\color{#c00}{17},22,26 \pmod{\!30}$.

So, $\!\bmod 30,\,$ any sequence $\,F_n\,$ satisfying said $\rm\color{#0a0}{recurrence}$ remains constant afterwards once it takes the value of one of those roots, e.g. your Fermat numbers, where $\,F_2\equiv \color{#c00}{17}\pmod{\!30}.$

Remark $ $ The same method works to solve for (modular) fixed points of any polynomial iteration, i.e. if the $\,a_i$ satisfy a recurrence $\,a_{n+1} = f(a_n)\,$ for a polynomial $\,f(x),\,$ then $\,a_n\,$ is a fixed point, i.e. $\, a_{n+1} = a_n\iff f(a_n) = a_n\iff a_n\,$ is a root of $f(x)-x,\,$ so finding fixed points of polynomial iterations reduces to finding roots of polynomials.

In your case note that subtracting $1$ from the roots shows they are roots of $\,x^2\equiv x,\,$ i.e. idempotents, so are $\equiv 0$ or $1$ for each prime modulus. Idempotents are well-studied since they play crucial roles in factorization of rings (e.g. they essentially govern the ring factorizations given by CRT = Chinese Remainder Theorem).

Answer 2

Of course $$F_n=F_0F_1\cdots F_{n-1}+2\equiv2\pmod{F_0F_1}$$ for $n\gt1$, and $F_n$ is odd, so there you have it.

Likewise, $F_n\equiv2\pmod{17}$ for $n\gt2$, $F_n\equiv2\pmod{257}$ for $n\gt3$, etc. This is why the Fermat numbers are pairwise relatively prime.