Unverified native note on distinct consecutive totient values

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Target and source identity

The target is Problem 1004: for every fixed $c>0$ and all sufficiently large $x$, find a consecutive block of length $(\log x)^c$ below $x$ on which Euler's totient values are pairwise distinct.

The five-page PDF (178,766 bytes), not held here because no license on record permits its redistribution, is titled A partial result on distinct consecutive values of Euler's function. It is the file linked from forum post 6051 of the site's discussion of Problem 1004, by the account aditya, posted at 18:09 on 29 April 2026 (the page's own clock; the page read 2026-09-07). The post links a Google Drive view address, https://drive.google.com/file/d/1qNREx0WwSBi3P4FYCglifN_D78p5zelH/view?usp=sharing, from which the file was downloaded. The post opens "Gpt 5.5 pro found a partial result"; that attribution is the post's. No author, venue or date is visible in the document, and its embedded creation time, 29 April 2026, is not treated as a publication date. The note has no stable bibliographic identity and no recorded review or acceptance.

Claims recorded from the note

Theorem 1 on local p. 1 claims that if $L=L(x)$ is positive, integer-valued, and

$$L\log(2L)=o((\log x)^2),$$

then, for all but $o(x)$ integers $n\leq x$, the values

$$\phi(n+1),\phi(n+2),\ldots,\phi(n+L)$$

are pairwise distinct. Corollary 1 specializes this to a claimed affirmative answer for every fixed exponent $c<2$ (in the positive-$c$ setting of Problem 1004). Corollary 2 gives the more explicit sufficient range

$$L\leq\frac{(\log x)^2}{(\log\log x)^{1+\delta}}$$

for each fixed $\delta>0$.

These are claims of an anonymous, unpublished note. They remain unverified. In particular, the note says nothing that resolves the target for $c\geq2$, and its own $c<2$ claim is not recorded as an established theorem or accepted solution. Problem 1004 remains open.

Inputs and proposed proof route

The note writes the shifted collision count as $P(X;k)=P_0(X;k)+P_1(X;k)$, following Graham--Holt--Pomerance and Pollack--Pomerance--Treviño. Graham--Holt--Pomerance Theorem 2 has only a fixed-$k$ eventual range. The uniform estimates actually quoted as Proposition 1 in the note correspond to Pollack--Pomerance--Treviño Theorem 3.1 and Theorem 3.3. The fixed-$k$ GHP theorem does not supply those growing-shift quantifiers and does not validate this note.

On local pp. 2--3 the note proposes the additional average estimate

$$\sum_{k\leq L}c(k)\ll\log L.$$

On local pp. 4--5 it lets $B(x,L)$ count starting points for which at least one collision occurs and uses

$$B(x,L)\leq\sum_{h=1}^{L-1}(L-h)P(x+L;h).$$

It then inserts the two uniform PPT13 estimates and its average bound to claim

$$B(x,L)\ll x\frac{L\log(2L)}{(\log x)^2} +\frac{xL^2}{\exp((\log x)^{1/3}/2)}=o(x).$$

This paragraph is a source-level proof pointer only. No estimate, summation, uniformity transition, or endpoint convention has been independently checked as a complete argument here.

Obstacles and next investigation

The source lacks a stable author/publication identity and any recorded peer review, acceptance, or independent proof verification. Its average bound for $c(k)$ and the transfer from individual shifted-collision estimates to an almost-all block statement need line-by-line mathematical review. The edition-specific PPT13 inputs and the fixed-$k$ limitation of GHP99 must remain explicit during that review.

A later investigation should verify the definition and convergence argument for $c(k)$, every uniform range and implied constant, the $x+L\leq2x$ passage, the treatment of odd shifts, and the final $o(x)$ estimates. It should also seek stable public provenance and independent acceptance evidence. Failure of this particular route would not refute Problem 1004, and checking it is not an authorization to solve the remaining $c\geq2$ cases.

Current review state

This lead is a research-only preservation record. Source identity, five-page layout, displayed claims, cited-input trace, and proof architecture were read. Independent source-fidelity and statement-scope reviews of this record are reported (2026-09-07), but their reports are not retained in this repository, so the record stands as author-recorded; no review of the note's argument exists. No full proof, formal verification, community acceptance, or change to the problem's open status is claimed. The native $c<2$ claim remains unverified.