On Locally Repeated Values of Certain Arithmetic Functions, I

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corollary_p320: Each of n+nu(n)=m+nu(m), n+Omega(n)=m+Omega(m) and n+tau(n)=m+tau(m), with n different from m, has infinitely many solutions.

theorem_1: Gives conditions on the average order and the maximum of a positive integer-valued arithmetic function f under which n+f(n)=m+f(m) has infinitely many solutions with n different from m.

theorem_2: Records the quantitative lower bound for collisions of n plus the number of distinct prime factors of n, a contextual result rather than a totient-block theorem.

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Paul Erdős, András Sárközy, and Carl Pomerance, On Locally Repeated Values of Certain Arithmetic Functions, I, Journal of Number Theory 21(3) (1985), 319--332, DOI 10.1016/0022-314X(85)90059-9.

Copy read. The copy read for this card is the published-article scan, with fourteen physical pages corresponding exactly to printed pp. 319--332. Its acquisition time is unknown. The PDF's 2002 scan metadata is not an acquisition date or a publication date. The scan prints "Reprinted from JOURNAL OF NUMBER THEORY" over "All Rights Reserved by Academic Press, New York and London" at the head of its first page, beside "Vol. 21, No. 3, December 1985", and "Copyright © 1985 by Academic Press, Inc. All rights of reproduction in any form reserved." at its foot, read on the page image because the text layer garbles these lines, every other right reserved.

The paper studies collisions of the map $n\mapsto n+\nu(n)$, where $\nu(n)$ is the number of distinct prime factors of $n$. Theorem 1, printed p. 320, gives hypotheses on the average order and the maximum of a positive integer-valued arithmetic function $f$ under which $n+f(n)=m+f(m)$ has infinitely many solutions with $n\ne m$. Its unnumbered corollary, on the same page, applies this to $\nu$, $\Omega$, and the divisor-counting function. Theorem 2 gives a quantitative lower bound for the number of collisions of $n+\nu(n)$, stated on printed p. 322 and proved in Sections 3 and 4, printed pp. 325--332.

The introduction, printed p. 320 (physical p. 2), announces later papers in the series and says that the authors will also obtain an upper bound for the number of solutions of $n+\nu(n)=m+\nu(m)$ and of $\phi(n)=\phi(n+1)$; no such bound is proved here. This source itself proves no theorem about a block of pairwise distinct totient values and does not address the every-$c$ target in Problem 1004. It is retained as exact historical context for the locally repeated-values series, not as direct progress on that problem.

Bears on. #1004 (adjacent context only: the problem page cites Theorem 2 and the introduction's announcement of later totient work to set them apart from the totient-block question; the paper proves nothing on that question).

Results to transcribe.

• Theorem 1 (printed p. 320): general sufficient conditions for infinitely many collisions of $n+f(n)$.
• Corollary (printed p. 320): the application to $\nu$, $\Omega$, and the divisor-counting function.
• Theorem 2 (printed p. 322): a quantitative lower bound for collisions of $n+\nu(n)$.

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