Problem 1003

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Statement. Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function?

Status. Open.

Source. erdosproblems.com/1003, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1003, https://www.erdosproblems.com/1003.

References.

• [EPS87] Erdős, Paul and Pomerance, Carl and Sárközy, András, On locally repeated values of certain arithmetic functions. III. Proc. Amer. Math. Soc. (1987), 1-7.
• [Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.

Formalization. Statement in formal-conjectures.

Current assessment

The standing judges the dated catalog formulation above.

A bounded status search checked primary arXiv records, author paper routes, searches for consecutive equal totients, and indexed X announcements. It found Li v2 still listed as the latest version and no primary proof settling unit-shift infinitude. This sampled search does not establish an exhaustive survey or independent acceptance of Li's proof. Direct catalog retrieval returned HTTP 403, so its 2026-09-04 statement is retained without claiming a fresh site reading.

Complete relevant pages were inspected: Erdős 1985 printed p. 67 / physical p. 3; Part II pp. 251--253 and its proof outline on pp. 257--258; Graham local pp. 1, 7--8; Kinlaw--Kobayashi--Pomerance local pp. 1, 11, 23--24 and abstract-only physical p. 25; Ford arXiv v5 pp. 1--2, 4; and Li v2 pp. 1--4, 35. These readings support statement, version, citation, and proof-location checks; they do not supply a full source-proof reconstruction or independent acceptance. In particular, the intermediate estimates and computational enumeration in the reciprocal-sum proof were not audited.

Li's separate Section 1.1 formalization claim asserts Lean 4 verification of the paper's results. No clone, build, declaration inspection, dependency audit, or axiom audit was performed here. The catalog's formal-conjectures statement is likewise not proof coverage for infinitude. The unresolved mathematical question and the outstanding full-proof reviews remain separate.

Progress

Put $S(x)=\#\{n\leq x:\phi(n)=\phi(n+1)\}$. Infinitude remains unresolved in the sources checked. The strongest unit-shift upper bound among them is Li's Corollary 1.5 in arXiv:2606.23681v2 (12 August 2026), which states

$$S(x)\ll x\exp\left\{-\left(\frac12-o(1)\right) \sqrt{\log x\log_2x}\right\},\qquad \log_2x=\log\log x.$$

This is a preprint result, not an infinitude theorem. Its proof on p. 35 specializes Theorem 1.4 and equation (1.4) to $h=1$. That theorem separates the shifted solution count into a same-prime-support diagonal and an error over its stated growing shift range. The diagonal is empty for odd $h$, so in particular for $h=1$. The analytic estimates behind the theorem have not been fully reconstructed or independently reviewed here.

Known Results

Erdős's 1985 survey, printed p. 67 / physical p. 3, records the expectation that $\phi(n)=\phi(n+1)=\cdots=\phi(n+k)$ has infinitely many solutions for every fixed $k$, while describing even the $k=1$ case as unattackable.

Erdős, Pomerance, and Sárközy's Theorem 2 in Part II (1987), printed p. 253 / physical p. 3, gives the historical bound

$$S(x)\leq x\exp\{-(\log x)^{1/3}\}$$

for sufficiently large $x$. Their following conjecture predicts at least $x^{1-\varepsilon}$ solutions for every $\varepsilon>0$ and sufficiently large $x$, while explicitly recording that infinitude was unknown. This is the direct totient source. The catalog bibliography's Part III reference must not be assigned Part II's p. 253 theorem; Part II's p. 251 unknown-infinitude sentence instead concerns the number of distinct prime factors.

Graham, Holt, and Pomerance's Theorem 2 bounds the solutions outside their parametrized family by $x\exp\{-(\log x)^{1/3}\}$ for $x\geq x_0(k)$, separately for each fixed shift $k$. For odd $k$ the family is empty, so $k=1$ gives an upper bound for all solutions. The selected 15-page author manuscript is dated 21 October 1997; the theorem and proof use local pp. 7--8. The 1999 publication span 867--882 is a separate page system. Neither this upper bound nor its $k$-dependent threshold proves infinitude or gives uniform control of shifts growing with $x$.

Li v2, equation (1.1) on local p. 2, reports Yamada's published improvement for the unit shift in the form

$$S(x)\ll x\exp\left\{-\left(2^{-1/2}+o(1)\right) \sqrt{\log x\log_3x}\right\},\qquad \log_3x=\log\log\log x.$$

The Kinlaw--Kobayashi--Pomerance paper also recalls Yamada's square-root improvement on local p. 1; its reference [11] on local p. 24 identifies T. Yamada, On equations $\sigma(n)=\sigma(n+k)$ and $\varphi(n)=\varphi(n+k)$, J. Combin. Number Theory 9 (2017), 15--21. This is a reported theorem from those two sources: Yamada's paper itself has not been read, extracted, or independently reviewed here. Thus the historical cube-root bounds above are not presented as the strongest published progress.

Kinlaw, Kobayashi, and Pomerance's Theorem 1.1 (2020) proves

$$\sum_{\substack{n\geq1\\\phi(n)=\phi(n+1)}}\frac1n<7.8358.$$

Their proof uses the reported exhaustive list of 10,755 solutions through $10^{13}$ and estimates for the remaining ranges. A finite enumeration and convergence of this sum allow either finite or infinite solution sets. The selected Online First copy has article pp. 1--24 and an abstract-only physical p. 25; the theorem is on local p. 1 and its proof occupies local pp. 11--23. Final issue pp. 69--92 are not locators for this copy.

Ford's Theorem 1 proves infinitude for every positive shift divisible by $442720643463713815200$. It also proves that there exists one fixed, unidentified even integer $\ell\leq3570$ such that, for every positive integer $k$ divisible by that $\ell$, the equation has infinitely many solutions. These are even-shift results and do not include $k=1$. The selected source is Solutions of $\phi(n)=\phi(n+k)$ and $\sigma(n)=\sigma(n+k)$, arXiv:2002.12155v5: the arXiv v5 stamp is 14 August 2020, its PDF footer says 17 August 2020, and the journal publication is IMRN 2022(5), 3561--3570. Theorem 1 is on local p. 2, with its proof on p. 4.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• banks_2018_counting_integers_smooth_totient
• banks_2018_counting_integers_smooth_totient / theorem_1_1
• erdos_1985_problems_results_number_theory
• erdos_1987_locally_repeated_values_certain_arithmetic_functions
• erdos_1987_locally_repeated_values_certain_arithmetic_functions / intro_phi_bound_p1
• erdos_1987_locally_repeated_values_certain_arithmetic_functions_ii
• erdos_1987_locally_repeated_values_certain_arithmetic_functions_ii / phi_sigma_conjecture_p253
• erdos_1987_locally_repeated_values_certain_arithmetic_functions_ii / theorem_2
• ford_2020_solutions_phi_n_phi_n_k
• ford_2020_solutions_phi_n_phi_n_k / theorem_1
• ford_2020_solutions_phi_n_phi_n_k / theorem_2
• graham_1999_solutions_phi_n_phi_n_k
• graham_1999_solutions_phi_n_phi_n_k / theorem_2
• kinlaw_2020_equation_phi_n_phi_n_1
• kinlaw_2020_equation_phi_n_phi_n_1 / theorem_1_1
• li_2026_rank_amplification_shifted_equal_values_euler_totient_function
• li_2026_rank_amplification_shifted_equal_values_euler_totient_function / corollary_1_5
• li_2026_rank_amplification_shifted_equal_values_euler_totient_function / formalization_claim_section_1_1
• li_2026_rank_amplification_shifted_equal_values_euler_totient_function / theorem_1_4