---

Claim. Theorem 3.1 of Quanyu Tang's note A Note on Erdős Problem #479: Infinitude of the Sets $A(2^i)$ and Related Results (dated 2 December 2025, posted in the author's GitHub repository and announced on the problem's forum thread the same day) proves that for every integer $i\ge1$ there are infinitely many positive integers $n$ with $2^n\equiv2^i\pmod n$. The construction takes $n=ip$ for odd primes $p\nmid i$ with $p\equiv1\pmod L$, where $L$ is the least common multiple of the orders of $2$ modulo the odd prime-power factors of $i$, each divided by its greatest common divisor with $i$; Dirichlet's theorem supplies infinitely many such primes. The problem page Problem 479 gives the construction in full.

Submission note. Posted to the site's forum by Quanyu Tang on 2 December 2025:

This website states that “Erdős and Graham report that Graham, Lehmer, and Lehmer have proved this for $k = 2^i$ for $i \ge 1$, or if $k = -1$, but I cannot find such a paper.” I have also tried to locate this manuscript, but without success.

Fortunately, I have just written a short note in which I give a complete proof of the case $k = 2^i$ for all $i \ge 1$, and survey the existing results for the set

$$A(k) := \{ n \ge 1 : 2^n \equiv k \pmod n \},$$

recording all

integers $k$ for which $A(k)$ is currently known to be infinite. In short, this problem remains open for every fixed $k$ other than

$$k \in \{0,-1,-2, 2^i > : i \ge 1\}.$$

This note is available on my GitHub page (here).

(The site has been updated to address this comment.)

Covers. The cases $k=2^i$ for every $i\ge1$. Other values of $k$ are not addressed by the theorem.

Depends on. Theorem 3.1 of Tang's note.

Standing. Tang disclaims novelty and says that the argument may or may not match the unlocated proof of Graham, D. H. Lehmer and Emma Lehmer that Erdős and Graham report. The note is unpublished, and the site labels the problem OPEN, so its commentary crediting the note with a proof for this case is not acceptance; the claim is claimed.