On the solutions to phi(n)=phi(n+k)

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theorem_2: Bounds solutions of phi(n)=phi(n+k) outside the paper's parametrized family, with an eventual threshold depending on the fixed shift k.

theorem_4: Gives equal totients along a finite arithmetic progression when associated linear forms are simultaneously prime.

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S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to $\phi(n)=\phi(n+k)$, in Number Theory in Progress, vol. 2, K. Győry, H. Iwaniec, and J. Urbanowicz, eds., de Gruyter, Berlin and New York, 1999, 867--882, doi:10.1515/9783110285581.867.

Copy read. The copy read for this card is the 15-page author manuscript, dated 21 October 1997 internally and using manuscript pp. 1--15; the final publication has bibliographic pp. 867--882. No page-by-page conversion between those locator systems is inferred. The manuscript was retrieved from https://math.dartmouth.edu/~carlp/phi.pdf during 2026-09-07T12:49:14.599601Z--2026-09-07T12:49:15.139899Z. Carl Pomerance's official publication page separately links a file named ghp.pdf; the two URL responses have not been shown to be byte-identical. That manuscript comes from the author's page (https://math.dartmouth.edu/~carlp/), which states no terms for the papers it links, and the manuscript prints no notice; the term is unstated.

The paper studies the count

$$P(k;x)=\#\{n\leq x:\phi(n)=\phi(n+k)\}$$

for a fixed shift $k$. Theorem 1 gives a parametrized family of solutions for even $k$. The authors split $P(k;x)=P_0(k;x)+P_1(k;x)$ into solutions of that form and the remaining solutions. Theorem 2 gives an unconditional upper bound for $P_1(k;x)$, separately for each fixed $k$. Corollary 1 gives a conditional asymptotic formula for $P(k;x)$ when $k$ is even, assuming the paper's quantitative prime-tuples Conjecture 2. Theorem 3 instead studies the sum of the structured counts over all $k\leq x$.

Theorem 4 constructs an arithmetic progression of equal totient values when a finite collection of associated linear forms are all prime. Corollary 2 obtains arbitrarily long such progressions only under the paper's prime-tuples Conjecture 1. Equal-totient progressions are distinct from the pairwise-distinct consecutive blocks sought in Problem 1004.

For Problem 1003, the $k=1$ instance of Theorem 2 is directly relevant quantitative information, but it does not prove that any, much less infinitely many, solutions exist. For Problem 1004, Theorem 2 has a $k$-dependent threshold and supplies no uniform estimate for a growing family of shifts. It therefore does not by itself justify a growing-shift union bound or prove pairwise-distinct totient blocks of length $(\log x)^c$ for $c<2$.

Source: https://math.dartmouth.edu/~carlp/.

Bears on. #1003; #1004 (fixed-shift collision bound and non-transfer context only).

Results to transcribe.

• Theorem 1: a parametrized family of equal-totient solutions for even shifts.
• Theorem 2: the eventual exceptional-solution bound for each fixed shift.
• Corollary 1: a conditional fixed-even-$k$ asymptotic under Conjecture 2.
• Theorem 3: $\sum_{k\leq x}P_0(k;x)\sim Cx$ for a finite positive constant defined in the source.
• Theorem 4 and Corollary 2: a finite-primality construction and its prime-tuples-conditional equal-totient progression consequence.

Living verification. Needs review. The identity, manuscript/final-page distinction, Theorems 2 and 4, and their proof pointers were checked against that author manuscript. No complete proof is supplied, reconstructed, or independently certified here.

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