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Source. Lemma 2, p. 357, of Yong-Gao Chen, On integers of the form $k2^n+1$, Proceedings of the American Mathematical Society 129(2), 355--361 (electronically published 28 August 2000), https://doi.org/10.1090/s0002-9939-00-05916-5, the edition named on the source card. A weak form with an ineffective constant is proved in Section 4, part (II), p. 360.

Read depth. Claims checked: the statement was read clause by clause on the printed page; the proof (p. 357) and the weak form (p. 360) were read for structure only. Nothing here is independently reviewed.

Statement

Lemma 2 (p. 357). "Let $p_1,\cdots,p_t$ be distinct odd primes and $x\geq3$. Then the number of positive odd integers $M\leq x$ such that there exist a positive integer $n$ and distinct primes $q_1,\cdots,q_r\in\{p_1,\cdots,p_t\}$ with

$$M2^n+1=q_1^{\beta_1}\cdots q_r^{\beta_r},\quad \beta_i\geq1,\quad i=1,\cdots,r,$$

is less than $c_1(\log\log x)(\log x)^r$, where $c_1$ depends only on $r$ and $p_1,\cdots,p_t$."

The constant is effective (p. 355).

Weak form (Section 4 (II), p. 360). For distinct primes $q_1,\ldots,q_r$ and $1\le M\le x$ with $M2^n+1=q_1^{\beta_1}\cdots q_r^{\beta_r}$, $n\ge1$, $\beta_i\ge0$: those with $n<\log x$ number fewer than $c^{(6)}(\log x)^{r+1}$, and those with $n\ge\log x$ satisfy $M\le c^{(8)}$, where every $c^{(i)}$ depends only on $q_1,\ldots,q_r$. The bound $c^{(8)}$ comes from the Mahler--Ridout theorem (Lemma 3, p. 359), and the paper calls the constants of Section 4 noneffective (p. 355). The paper says this weak form suffices for its purpose.

Proof pointer

Proof on p. 357. For fixed $q_1,\ldots,q_r$, Yu's bound (Lemma 1, p. 357) applied to $q_1^{\beta_1}\cdots q_r^{\beta_r}-1=M2^n$ gives $n\le c\log(12\max\beta_i)$, equation (2); comparing sizes then gives $\max\beta_i<c_2\log x$, equation (3), so $n\le c_3\log\log x$, and counting the choices of $n$ and the $\beta_i$ gives the bound, summed over the $\binom{t}{r}$ choices of primes. The weak form replaces Lemma 1 by the Mahler--Ridout theorem (Lemma 3, p. 359).

Dependencies

Lemma 1 (p. 357), a special case of the corollary of Theorem 1 of K. Yu, Linear forms in $p$-adic logarithms, III, Compositio Math. 91 (1994); for the weak form, Lemma 3 (p. 359), from Mahler (1957) and Ridout (1957).

Bears on

• Problem 1113: the paper does not mention the problem. The lemma counts coefficients $M\le x$ relative to a prime set fixed in advance; it says nothing about the covering sets of any single coefficient, so it neither produces a Sierpiński number without a finite covering set nor rules one out.