Ford 2020 solutions phi n phi n k

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theorem_1: Gives unconditional infinitude for every multiple of an explicit even modulus and for every multiple of some even shift at most 3570.

theorem_2: For each m at least 3 gives distinct offsets h_1,...,h_m such that, for every natural number l, the totients at n+l*h_j all agree for infinitely many n.

theorem_4: Shows that sigma(n)=sigma(n+k) has infinitely many solutions n for a positive proportion of all natural numbers k, without naming any such k.

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Kevin Ford, Solutions of $\varphi(n)=\varphi(n+k)$ and $\sigma(n)=\sigma(n+k)$, International Mathematics Research Notices 2022 (2022), no. 5, 3561--3570, DOI 10.1093/imrn/rnaa218; published online 26 August 2020.

Copy read and date guard. The copy read for this card is arXiv:2002.12155v5, with six physical and numbered pages. Its visible arXiv watermark says 14 August 2020, while its manuscript footer is separately dated 17 August 2020. Neither date is relabeled as the other. Locators below refer to this arXiv copy, not to final journal pagination. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2002.12155), every other right reserved.

Write $\mathcal S_k$ for the hypothesis that $\varphi(n)=\varphi(n+k)$ holds for infinitely many $n$. Theorem 1 proves two unconditional even-shift results:

• $\mathcal S_k$ holds whenever $442720643463713815200\mid k$.
• For some even $\ell\leq3570$, $\mathcal S_k$ holds for every multiple $k$ of $\ell$.

Literal source wording. Theorem 1(b), on p. 2, continues: “consequently, the number of $k\leq x$ for which $\mathcal S_k$ is true is at least $x/3570$.”

Compiler qualification. That floor-free pointwise consequence has a rounding imprecision. For real $x\geq0$, the substantive divisibility clause gives the exact lower bound

$$\#\{k\leq x:\mathcal S_k\text{ is true}\} \geq\left\lfloor\frac{x}{\ell}\right\rfloor \geq\left\lfloor\frac{x}{3570}\right\rfloor.$$

In particular, the set of such shifts has lower natural density at least $1/\ell\geq1/3570$. This is a compiler qualification of the printed consequence, not an author-issued erratum or a separate theorem.

The proof combines Lemma 3 (p. 4), which turns a pair of simultaneous prime forms $ar+1$, $br+1$ into $\mathcal S_k$ for every multiple of an explicit even number $\kappa(a,b)$, with Lemma 2 (p. 2), the paper's form of recent progress toward the prime $k$-tuples conjecture due to Zhang, Maynard, Tao, and Polymath. On p. 4 it chooses two explicit 50-element sets, computes the relevant least common multiple and maximum as $442720643463713815200$ and $3570$, and applies Lemma 3.

The paper also proves, unconditionally, Theorem 2 (p. 2), a simultaneous equal-totient result along dilations of one unspecified tuple of offsets, and Theorem 4 (p. 3), that $\sigma(n)=\sigma(n+k)$ has infinitely many solutions for a positive proportion of $k$. Theorems 3 and 5 (p. 3) are conditional and have no pages here: Theorem 3 assumes $\mathrm{DHL}^*(5;2)$ or $\mathrm{DHL}^*(4;2)$, and Theorem 5 assumes $\mathrm{DHL}^*(t;m)$ together with integers $a_1,\dots,a_t$ sharing one value of $\sigma(a)/a$.

For Problem 1003, Ford's theorem is adjacent-shift context only. Every shift produced by Theorem 1 is even. The source itself observes in the Lemma 1 setup that the necessary same-prime-factor pair cannot occur for odd $k$. Thus no result here transfers to the unit shift $k=1$. Theorem 2 leaves its offsets unspecified, so it too gives no statement for $k=1$. The prior source annotation that the E1003 forum cited Ford alongside the Schinzel and Tao observation that shift $2$ is more tractable than shift $1$ is retained as contextual provenance, not as a unit-shift theorem.

Source: https://arxiv.org/abs/2002.12155.

Bears on. #1003, context only: Theorem 1 gives infinitude for even shifts only, and Theorem 2 for unspecified offsets; neither gives the unit shift $k=1$.

Results to transcribe.

• Theorem 1 (p. 2): infinitude for every multiple of the displayed even modulus and for every multiple of some even $\ell\leq3570$.
• Theorem 2 (p. 2): for each $m\geq3$, distinct offsets $h_1,\dots,h_m$ such that, for every $\ell\in\mathbb N$, $\varphi(n+\ell h_1)=\dots=\varphi(n+\ell h_m)$ for infinitely many $n$.
• Theorem 4 (p. 3): infinitely many equal divisor sums for a positive proportion of shifts.

Living verification. Needs review. The selected-version identity, two distinct visible dates, the exact statements of Theorems 1, 2 and 4, and their proof pointers were checked against arXiv v5. The pre-existing E1003 relationship is preserved with its even-shift limitation explicit. 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.