Doorn 2026 three term arithmetic progressions consecutive powerful

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conjecture_5: Conjectures that the consecutive powerful progressions of the shape (x-2)^2, (x-1)^2, 7^3 y^2 = x^2-2 in [1, n] number (C_7 + o(1)) log n with an explicit constant C_7 of about 0.0014.

conjecture_7: Conjectures that for each i in {0, 1, 2} the consecutive powerful progressions up to n containing exactly i squares number of order log n, so that there are infinitely many in all.

corollary_3: Gives a sufficient condition, in terms of fractional parts of the Pell sequence x_k over every squarefree m other than 1 and 7, for infinitely many of the Theorem 1 triples to be consecutive powerful numbers.

lemma_2: Shows that for every integer x at least 3 the powerful numbers strictly between (x-2)^2 and x^2 are counted by the squarefree m with the fractional part of x/m^{3/2} below 2/m^{3/2}.

lemma_6: Shows that a three-term progression of consecutive powerful numbers contains exactly two squares if and only if it is (x-2)^2, (x-1)^2, x^2-2 for some x at least 3.

theorem_1: Proves that infinitely many three-term arithmetic progressions N, N+d, N+2d of powerful numbers have common difference d equal to 2√N + 1.

theorem_4: Proves that for each fixed squarefree m at least 648560 the fractional-part inequality of Corollary 3 holds for infinitely many terms of the Pell sequence x_k.

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Wouter van Doorn, Three-term arithmetic progressions of consecutive powerful numbers. arXiv preprint (2026). arXiv:2605.06697.

The copy read for this card is arXiv:2605.06697v1 [math.NT], dated 4 May 2026, ten pages; the pages cited below are that version's.

Theorem 1 (p. 2) gives infinitely many N for which N, N+d, N+2d with d = 2√N + 1 are all powerful, sharpening Chan's unconditional d <= 4√N + O(1); the construction is Pellian, taking solutions of x^2 - 7^3 y^2 = 2 and using the triple (x-2)^2, (x-1)^2, 7^3 y^2 = x^2 - 2, with infinitude supplied by solving x^2 - 7y^2 = 2 and restricting to 7 | y, which happens exactly for the indices k ≡ 3 (mod 7) (Section 3.2, pp. 2--3). Section 4 studies when such a progression is made of three consecutive terms of the powerful-number sequence: Lemma 2 (p. 4) counts the powerful numbers strictly between (x-2)^2 and x^2 through fractional parts of x/m^{3/2} over squarefree m, and Corollary 3 (p. 5) turns it into a sufficient condition on the recurrence terms x_k. Theorem 4 (p. 5) shows that for each fixed squarefree m >= 648560 the corollary's inequality holds for infinitely many k, one modulus at a time and not for all m together, and a heuristic density computation leads to Conjecture 5 (p. 6), which predicts (C_7 + o(1)) log n consecutive triples of this shape in [1, n], with C_7 about 0.0014. Lemma 6 (p. 7) shows that a progression of consecutive powerful numbers contains exactly two squares if and only if it has the shape (x-2)^2, (x-1)^2, x^2-2 for some x >= 3, so that it comes from a Pell equation. Conjecture 7 (p. 8) predicts that, for each i in {0, 1, 2}, the progressions of consecutive powerful numbers up to n containing exactly i squares number of order log n. Section 5.3 (pp. 8--9) reports a search below 10^14 that finds 18 such progressions, every one with exactly one square, so none of the Theorem 1 shape. Bearing on #938: Erdős asked whether there are only finitely many three-term progressions of consecutive powerful numbers, and the paper says its d = 2√N + 1 reaches the threshold relevant to that question (p. 1). The negative answer to Erdős's question remains conjectured, not proved. The AI disclosure (Section 2, pp. 1--2) states that ChatGPT served as a sounding board, for proofreading and for numerical examples, among them the calculation of the recurrence (4) and Table 1, and that "The paper itself was entirely human-generated." (p. 2).

Read status. Claims checked: Theorem 1, Lemma 2, Corollary 3, Theorem 4, Conjecture 5, Lemma 6 and Conjecture 7 were read clause by clause against the print (pp. 1--9). The proofs of Theorem 1, Lemma 2 and Lemma 6 were read; the proof of Theorem 4 rests on an external theorem of Chen, Ye and Zheng that was not checked here.

Source: https://arxiv.org/abs/2605.06697. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2605.06697), every other right reserved.

Results

• Theorem 1 (p. 2)
• Lemma 2 (p. 4)
• Corollary 3 (p. 5)
• Theorem 4 (p. 5)
• Conjecture 5 (p. 6)
• Lemma 6 (p. 7)
• Conjecture 7 (p. 8)

Bears on. #938: Theorem 1 constructs progressions of powerful numbers that are candidates for consecutive ones, Corollary 3 gives a sufficient condition for infinitely many of them to be consecutive, and Lemma 6 shows that every consecutive progression with two squares has the shape (x-2)^2, (x-1)^2, x^2-2 for some x >= 3, the shape of Theorem 1's triples. Conjectures 5 and 7, if true, would give infinitely many progressions of consecutive powerful numbers and so answer the question in the negative; they are not proved, and the paper does not decide the problem.

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