On powers of the diophantine function $\star:x\mapsto x(x+1)$
Donald Silberger
TL;DR
The paper studies the powers of the diophantine map $\star:x\mapsto x(x+1)$ by introducing the gross sequence $\gamma_\star(x)$ and the star sequence $x^\star$, revealing structure such as pairwise coprimality and prime-avoidance phenomena (there exists a prime $p(x)$ not dividing any $\star^k x+1$). It develops a universal embedding framework via a 'mother sequence' $m$ of prime-power factors and shows every sequence $\eta$ of prime powers has a parallel embedding into $m$, yielding infinitely many disjoint copies $c_j$ of $\eta$ within $m$. The paper also establishes a Sylvester-type exact sum $1/x=\sum_{k=0}^{\infty}1/(\star^k x+1)$ and proves the corresponding reciprocal series $\sum_{j} 1/x_j$ diverges for all $x$, combining Odoni's and Mertens' results with probabilistic reasoning. Collectively, these results connect diophantine constructions to Sylvester-type sequences, illuminate prime-distribution patterns in these towers, and extend the understanding of infinite decompositions and divergent harmonic-type sums in this setting.
Abstract
We treat the functions $\star^k:{\mathbf N}\rightarrow{\mathbf N}$ where $\star:x\mapsto \star x := x(x+1)$. The set $\{\star^k x+1: \{x,k\}\subseteq{\mathbf N}\}$ is pairwise coprime; so, the set ${\mathbf P}$ of primes is infinite. Our Theorem 4 resorts to the mother sequence, M, that is obtained by factoring the infinite sequence $2,3,4,5,\ldots$ into prime powers. For each $x\ge1$ we define the gross $x$-sequence, $γ_\star(x) := \langle x+1; \star x+1; \star^2x+1; \star^3x+1;\ldots\rangle$, and also the star sequence, $x^\star$, obtained by factoring the terms of $γ_\star(x)$ into prime powers. It turns out that $γ_\star(1)$ is Sylvester's sequence, A00058 in the Online Encyclopedia of Integer Sequences, OEIS, and that $γ_\star(2)$ is the sequence A082732 in the OEIS. Theorem 3. For every integer $x\ge1$ there is a prime $p(x)$ that divides no member of $\{\star^kx+1: k\ge0\}$. Theorem 4. For each sequence $η$ of powers of primes there are infinitely many subsequences $c$ of M such that numerically $η=c_j$ but where the term-set family in M of those $c_j$ is formally. pairwise disjoint. Theorem 6. $1/x = \sum_{k=0}^{n-1} 1/(\star^kx+1) + 1/(\star^nx) = \sum_{k=0}^\infty 1/(\star^kx+1)$ for all $\{x,n\}\subseteq{\mathbf N}$. Theorem 7. For every $x\in{\mathbf N}$, when $x^\star := \langle x_j\rangle_{j=0}^\infty$ then $\sum_{j=0}^\infty 1/x_j = \infty$.