The Binary Totient Series

Theorem 6.155 (A sufficient order hypothesis for unbounded prime support). Let $C:\mathbb{N}\to \mathbb{N}$, and fix integers $m\ge 1$ and $d\ge 0$. Suppose that for every pair of primes $q,p$ with $p\mid C(mq)$ there is an integer $k$ such that

The divisibility forces $gcd(p,mq)=1$. For such $p$, existence of $k$ in the displayed range is equivalent to ${\mathrm{ord}}_{mq}(p)\le d$. It also gives $mq<p^d$. Hence every fixed finite set of primes is disjoint from the prime divisors of $C(mq)$ for all sufficiently large prime $q$. If, in addition, for all sufficiently large prime $q$ one has $C(mq)>1$ and $gcd(C(mq),mq)=1$, then for every $B,N_0$ there are primes $q\ge N_0$ and $p>B$ with $p\mid C(mq)$.

Indeed, the divisibility gives $mq\le p^k-1<p^k\le p^d$. For a finite set of primes, take $q$ larger than all their $d$-th powers; for the last assertion, take a prime divisor of the nontrivial value $C(mq)$ after this threshold. The stated coprimality is included in the formal source’s layer hypothesis, although this last extraction argument uses only $C(mq)>1$. For $C(n)=2^n-1$, the first theorem above supplies the hypothesis with $m=d=1$: here ${\mathrm{ord}}_q(p)=1$, whereas ${\mathrm{ord}}_p(2)=q$. For $C(n)=|{\mathrm{\Phi }}_n(2)|$ and a general fixed $m>0$, the source uses the eventual version: the divisibility is required only for prime $q$ above a fixed threshold. The same proof then applies after that threshold. It is not valid to replace this by an all-prime assertion: ${\mathrm{\Phi }}_6(2)=3$, but $6\nmid 3-1$. For the binary cyclotomic family, choosing $q>max(m,2^m)$ removes these exceptional indices. Other families require their own proof of the divisibility and nontriviality assumptions.