Bajpai 2024 arithmetic progressions squarefull numbers

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accepted_manuscript_example: Records and directly verifies the revised explicit four-term squarefull progression in the later author manuscript.

corollary_1_3: Combines unconditional constructions with abc-based finiteness bounds to determine A-infinity of k conditionally.

evidence/: Exact integer certification of the Section 5 finite computations and the accepted manuscript's 190-digit example.

lemma_2_1: Bounds the radical of a k-full number after division by a k-full divisor.

lemma_2_2: Strengthens the radical estimate for at least 2k-1 consecutive k-full terms.

lemma_3_1: Packages consecutive progression terms into a three-term abc identity.

proposition_5_1: Computes the valuations that force opposite parity in the four-term squarefull construction.

proposition_5_2: Gives the elliptic-curve and congruence construction that resolves Erdos Problem 937 unconditionally.

theorem_1_1: Gives conditional lower bounds for the gcd and comparison bounds for the initial term and common difference of a k-full progression.

theorem_1_2: Constructs infinite primitive families for (m,k) equal to (3,2), (3,3), and (4,2), the last resolving Erdos Problem 937.

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Prajeet Bajpai, Michael A. Bennett, and Tsz Ho Chan, "Arithmetic progressions in squarefull numbers," International Journal of Number Theory 20 (2024), no. 1, 19--45. DOI. The accepted author manuscript PDF prints "© World Scientific Publishing Company" in its first-page running head and no license statement, every other right reserved. For the arXiv v1 PDF, the arXiv record names arXiv's non-exclusive distribution license (arXiv:2302.03113), every other right reserved.

The paper studies arithmetic progressions of $k$-full numbers, meaning positive integers in which every prime divisor occurs to exponent at least $k$. Its unconditional Theorem 1.2 constructs infinitely many primitive progressions for the exceptional pairs

$$(m,k)\in\{(3,2),(3,3),(4,2)\}.$$

The four-term construction is stronger than the theorem's stated $\gcd(N,d)=1$: the four squarefull terms are pairwise coprime. It therefore settles Erdos Problem 937. The construction uses a rational point on an elliptic curve, division polynomials modulo $73$, and a separate $2$-adic calculation to force the required congruence and parity conditions.

Theorem 1.1 is conditional on the $abc$ conjecture. It gives lower bounds for $\gcd(N,d)$ and comparison bounds between $N$ and $d$ in any progression of $k$-full numbers. The conditional bounds leave only finitely many primitive progressions of each longer length and, together with the unconditional constructions, imply

$$A^\infty(2)=4,\qquad A^\infty(3)=3,\qquad A^\infty(k)=2\quad(k\geq4).$$

The copy read for this card is the later accepted author manuscript; the two editions are:

• accepted author manuscript, June 26, 2023, 26 pages; author-hosted source.

arXiv v1, submitted February 6, 2023; manuscript dated February 8, 27 pages; arXiv:2302.03113v1.

The accepted manuscript replaces the introductory four-term example from v1 with a much smaller 190-digit example, changes the elliptic-curve point used to obtain it from $14P_1-8P_2+T_1$ to $2P_1-6P_2+T_2$, and credits Gary Walsh with finding the smaller example, which fixed a calculation slip by the second author (p. 25). The numbered theorems and their mathematical statements are otherwise stable between these versions. Result pages here cite the accepted manuscript by its printed page numbers, which coincide with its PDF page numbers.

The paper also proves, unconditionally, that squarefull progressions that need not be primitive can have a small common difference. For each $m\geq4$ it constructs infinitely many $m$-term progressions with $d\mid N$ and $d\ll N^{(2m-4)/(2m-3)}(\log N)^{-2/(2m-3)}$ (Theorem 6.2 and its proof, pp. 19--21). This gives the unconditional upper bound $\theta_m\leq(2m-4)/(2m-3)$ of Proposition 6.1, where $\theta_m$ is the lower limit of $\log d/\log N$ over $m$-term squarefull progressions. Section 7 notes that the least common difference $d_m$ of an $m$-term squarefull progression satisfies $d_m\leq\prod_{p\leq m}p^2$, the product over primes (p. 24). These results bound the smallest possible difference, not the difference in every progression, and they do not strengthen the coprime existence assertion of Problem 937.

Bears on. #937.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.