On the Equation Phi(n) = Phi(n+1)

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theorem_1_1: Proves that the reciprocal sum of the integers n with phi(n) = phi(n+1) is less than 7.8358.

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Paul Kinlaw, Mitsuo Kobayashi, and Carl Pomerance, On the Equation $\varphi(n)=\varphi(n+1)$, Acta Arithmetica 196 (2020), no. 1, 69--92, DOI 10.4064/aa190627-20-1; published online 15 June 2020.

Edition guard. The copy read for this card is the Online First PDF, not the final-paginated issue copy. It contains local article pp. 1--24 (physical pp. 1--24) followed by an abstract-only physical p. 25. The first article page visibly bears the local label [1]. The final issue span 69--92 is bibliographic metadata and is not used to locate claims in that copy. The original retrieval time of that copy is unknown. It prints "© Instytut Matematyczny PAN," followed by three asterisks in place of the year on its first page, and the publisher's record offers the PDF to subscribers only and carries no CC-BY label (https://www.impan.pl/get/doi/10.4064/aa190627-20-1), every other right reserved.

Put

$$\mathcal S=\{n\in\mathbb N:\varphi(n)=\varphi(n+1)\}.$$

Theorem 1.1, on local p. 1, proves

$$\sum_{n\in\mathcal S}\frac1n<7.8358.$$

The introduction, local p. 1, says that it is still not known whether $\mathcal S$ is infinite, and recalls that Bayless and Kinlaw (its reference [1]) had bounded the reciprocal sum by 441702 and conjectured that it is less than 2; Theorem 1.1 improves that upper bound. The proof uses the exact computation of $\mathcal S$ up to $10^{13}$, an averaging argument that limits the odd member of each pair, and, by local p. 2, further techniques for the range $n>e^{150}$; the count of 10,755 solutions in the computed range is given in Section 4.1, on local p. 11. A finite computation and a convergent reciprocal-sum bound do not decide infinitude.

Section 4, local pp. 11--23, proves Theorem 1.1 by splitting the reciprocal sum into the small range through $10^{13}$, a middle range through $X_0=e^{150}$, and the remaining large range. The last page combines the three bounds as $1.4325+3.8006+2.6027=7.8358$.

Problem 1003 asks whether $\mathcal S$ is infinite. The theorem bounds the reciprocal sum of $\mathcal S$ and is consistent with either answer; the paper proves neither infinitude nor finiteness, and its introduction states that the question is open.

Bears on. #1003 (the problem page cites Theorem 1.1, a bound on the reciprocal sum of the solutions of $\varphi(n)=\varphi(n+1)$; it does not decide whether there are infinitely many solutions).

Results to transcribe.

• Theorem 1.1 (local p. 1): the reciprocal sum over the solutions of $\varphi(n)=\varphi(n+1)$ is less than 7.8358.

Living verification. Needs review. The final bibliographic identity, selected-edition map, theorem statement, computation cutoff, and proof endpoints were checked against the adopted metadata and selected Online First PDF. No complete proof is supplied, reconstructed, or independently certified here.

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