Tang (2025): A note on Erdős Problem 479

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theorem_3_1: Tang's explicit proof that, for every integer i at least 1, infinitely many positive integers n satisfy 2^n ≡ 2^i (mod n), by taking n = ip for primes p in a progression given by multiplicative orders.

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Quanyu Tang's A Note on Erdős Problem #479: Infinitude of the Sets $A(2^i)$ and Related Results (2 December 2025) is an unpublished author manuscript. Its p. 1 explicitly disclaims novelty of the underlying number-theoretic statements and describes the power-of-two argument as expository and independent; it may or may not coincide with the unpublished Graham–Lehmer–Lehmer proof. No accepted or published version was located by the searches recorded on the problem page.

Theorem 3.1, stated on p. 4 and proved on pp. 4–5, states that for every integer $i\geq1$ there are infinitely many positive integers $n$ with $2^n\equiv2^i\pmod n$. Section 2 (pp. 1–3) surveys the values of $k$ for which $A(k)=\{n\ge1:2^n\equiv k\pmod n\}$ is known to be infinite, and ends (p. 3) by saying that the question remains open for every fixed $k$ other than $k\in\{0,-1,-2,2^i:i\geq1\}$.

Read status: claims checked for Theorem 3.1 (p. 4), read clause by clause on the page images with its proof (pp. 4–5) read step by step; the p. 1 disclaimer and the Section 2 survey (pp. 1–3) were read on the page images. Nothing is independently reviewed.

Bears on. #479: Theorem 3.1 proves the problem's assertion for the values $k=2^i$ with $i\geq1$ and for no other $k$. As the note reports on p. 1, Erdős and Graham (1980, p. 96) attribute to Graham, Lehmer and Lehmer the partial result for these $k$ and for $k=-1$; Theorem 3.1 covers its cases $k=2^i$ only. The Section 2 survey (p. 3) lists $k\in\{0,-1,-2\}$ as the other values for which infinitude is known: $k=0$ as immediate ($A(0)$ is the set of powers of $2$), $k=-1$ with Kalmynin and the OEIS, and $k=-2$ with Kin Y. Li et al. Paged at theorem_3_1.

Extracted result. Theorem 3.1 (p. 4): for every integer $i\geq1$, infinitely many positive integers $n$ satisfy $2^n\equiv2^i\pmod n$.

Edition read. The copy read for this card is the unpublished author manuscript dated 2 December 2025 on its p. 1, which prints no notice and no arXiv stamp. The manuscript is posted as A_note_on_Erdos_Problem_479.pdf in the author's GitHub repository (https://github.com/QuanyuTang/Erdos-Problem-479-Note), the posting linked from https://www.erdosproblems.com/479; read 2026-10-07, that GitHub repository holds the PDF, its TeX source and a README whose citation line gives the same date, and no license file. An arXiv query for the author and title on 2026-10-02 found no arXiv record for the paper; the term is unstated.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.