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Let $a_0 = 2^{2^k} + 1$ for $k$ sufficiently large, and let $$a_n = (a_{n-1} - 1)^2 + 1.$$ Then $a_n = 2^{2^{k+n}} + 1$, so this sequence can't always be prime regardless of the value of $k$ iff the …

As mentioned in the comments, there is no induction, because Euler's theorem isn't true for all values of $a$, only those coprime to $n$. Alternatively, here is a generalization of Fermat's little th …

The space of sequences satisfying a linear recurrence is closed under multiplication (exercise). In particular, $F_n^2$ is such a sequence. Or, more simply: every polynomial sequence itself satisfies …

In fact the following is true. Proposition: Let $\mathcal{O}_K$ be the ring of integers of a number field $K$. For any $p_0$, it is possible to find finitely many primes $p_1, ..., p_n \ge p_0$ such …

Not a complete answer: these sequences are called Beatty sequences. It's a nice exercise to show that $S(\alpha)$ and $S(\beta)$ partition the positive integers when $\frac{1}{\alpha} + \frac{1}{\beta …

Here are some naive comments. If $X, Y$ are large then the contribution of the quadratic terms swamps the other terms, so first let's concentrate on the quadratic terms $$Q(X, Y) = a X^2 + b XY + c Y^ …

No; in fact, you can take $K = \mathbb{C}$. This follows from the Mason-Stothers theorem in the same way that FLT for sufficiently large $n$ follows from the abc conjecture. As Chandan indicates in …

Theorem: Let $\alpha$ be an algebraic integer such that $\mathbb{Z}[\alpha]$ is integrally closed, and let its minimal polynomial be $f(x)$. Let $p$ be a prime, and let $\displaystyle f(x) \equiv …

The answer is no. As the Wikipedia article in my comment states, the counterexample $p = 17, q = 3313$ was found by Stephens in 1971, but the stronger question of whether one can ever divide the othe …

I'll assume you mean the natural logarithm. The condition that $\log \log m = q \log \log n$ where $q = \frac{a}{b}$ is rational is equivalent to the condition $(\log m)^b = (\log n)^a$. If $m \neq …

This is not even true for quadratic extensions. Given primes $p_1, p_2, ... p_n$ find a prime $q \equiv 1 \bmod 4p_1 ... p_n$, which exists by Dirichlet's theorem, and consider $K = \mathbb{Q}(\sqrt{ …

J. C. Ottem's example $1^2 + ... + 24^2 = 70^2$ in the comments is of particular mathematical interest; it is one way to construct the Leech lattice, and is therefore somehow mysteriously related to o …

And if you insist, let me write this out in detail. All you need is the following lemma. Lemma: Let f(n) be periodic with period p and let g be injective. Then g(f(n)) is periodic with period p. …

Generating functions is not really the right name. I would say "parameterization." These formulas were known to the Babylonians. The simple proof goes as follows: assume WLOG that a, b, c are relat …

I more or less agree with Kevin; "elementary" to me means "from first principles." Another way I would put this is that if Gauss didn't know it, it's not elementary.