On Locally Repeated Values of Certain Arithmetic Functions II

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phi_sigma_conjecture_p253: Records the authors' quantitative conjecture and explicit unknown-infinitude statement for consecutive equal totients and divisor sums.

theorem_2: Bounds the number of n up to x for which phi(n) equals phi(n+1) by x divided by the exponential of the cube root of log x.

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Paul Erdős, Carl Pomerance, and András Sárközy, On Locally Repeated Values of Certain Arithmetic Functions. II, Acta Mathematica Hungarica 49 (1987), no. 1--2, 251--259, DOI 10.1007/BF01956329.

Copy read. The copy read for this card is the published scan, with nine physical pages corresponding directly to printed pp. 251--259; its retrieval time is unknown. No notice is printed in the scan; the publisher's article page shows "© Akadémiai Kiadó 1987" and names no license (Springer article page for DOI 10.1007/BF01956329, read 2026-10-02), every other right reserved.

The opening two pages study the number of distinct prime factors $\nu(n)$ and related functions. In particular, the sentence on printed p. 251 saying that infinitude is unknown concerns $\nu(n)=\nu(n+1)$, not Euler's totient function.

The totient equation is introduced as (1.5) on printed p. 252. On printed p. 253, Theorem 2 proves that, for all sufficiently large $x$,

$$\#\{n\leq x:\varphi(n)=\varphi(n+1)\} \leq \frac{x}{\exp\{(\log x)^{1/3}\}}.$$

The same paragraph says the proof gives the analogous bound for $\sigma(n)=\sigma(n+1)$. The following unnumbered conjecture predicts at least $x^{1-\varepsilon}$ solutions up to $x$ for each equation, for every $\varepsilon>0$ and sufficiently large $x$, and states that the authors could not prove even infinitude for either equation.

Section 4, printed pp. 257--258, gives an outline of the proof of Theorem 2. It discards as negligible the integers failing four largest-prime-factor and power conditions, (i)--(iv), assumes as condition (v) that the largest prime factor of $n$ exceeds that of $n+1$ (the opposite case is treated the same way), then counts the remaining possibilities; the source explicitly says the argument is nearly the same as the amicable-number proof in its reference [8].

For Problem 1003, this is the direct 1987 source for both the unit-shift upper bound and the historical infinitude conjecture. For [[../wiki/problems/arithmetic_functions/E1004/_index|Problem 1004]], it is only adjacent unit-shift collision context: it neither states nor proves the long block of pairwise distinct totient values.

Source: https://www.renyi.hu/~p_erdos/1987-14.pdf.

Bears on. #1003; #1004 (context only).

Results to transcribe.

• Theorem 2: the $x/\exp\{(\log x)^{1/3}\}$ upper bound for consecutive equal totients, with the same bound for consecutive equal divisor sums.
• Unnumbered conjecture on p. 253: at least $x^{1-\varepsilon}$ solutions for each of the two equations, together with the authors' explicit inability to prove infinitude.

Living verification. Needs review. The identity, page map, exact Theorem 2 statement, following conjecture, and proof-section pointer were checked against that scan. No complete proof is supplied, reconstructed, or independently certified here.

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