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Chan 2025 note three consecutive powerful numbers
Chan, Tsz Ho, A note on three consecutive powerful numbers. Integers 25 (2025), Paper No. A7, 7 pp. doi:10.5281/zenodo.14679327.
A number n is powerful if p | n implies p^2 | n, and the Erdos-Mollin-Walsh conjecture (Conjecture 1 here) asserts no three consecutive powerful numbers exist. The note settles a special shape of the conjecture: Theorem 1 shows there are no triples x^3-1, x^3, x^3+1 of powerful numbers with x^3-1 = p^3 y^2 and x^3+1 = q^3 z^2 for primes p, q and positive integers x, y, z. Corollary 1 deduces that 64x^6 - 1 = p^3 q^3 y^2 has no integer solutions with p, q prime. The proof combines elementary factorization and 3-adic valuation lemmas on x^2 +/- x + 1, a reduction of quartic equations y^2 = ax^4 + cx^2 + e to elliptic curves Y^2 = X^3 + cX^2 + aeX, and solutions of the Pell equation x^2 - 3y^2 = 1 together with second-order recurrences. It gives partial progress on Erdos problem 364, which asks whether three consecutive integers can all be powerful, and notes that the abc-conjecture implies there are only finitely many such triples.
Source: http://math.colgate.edu/~integers/vol25.html. No notice is printed; the journal's home page states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License so that all content is freely available without charge to the users or their institutions." (https://math.colgate.edu/~integers/, read 2026-10-02): the Creative Commons Attribution 4.0 license.
Bears on. #364
Results to transcribe.
- Theorem 1: There are no primes p, q and positive integers x, y, z with x^3-1 = p^3 y^2 and x^3+1 = q^3 z^2, so no three consecutive powerful numbers x^3-1, x^3, x^3+1 have this shape.
- Corollary 1: The Diophantine equation 64x^6 - 1 = p^3 q^3 y^2 has no solution in integers x, y with p, q prime.
- Lemma 4: For fixed integers a != 0, c, e, an integer point with x nonzero on y^2 = ax^4 + cx^2 + e yields an integer point with X = ax^2 nonzero on the elliptic curve Y^2 = X^3 + cX^2 + aeX.