Prime-Power Transport and a Three-Site Reduction

Payam Danesh; Raoul Bianchetti

doi:https://doi.org/10.14445/22315373/IJMTT-V72I9P101

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 9 | Year 2026 | Article Id. IJMTT-V72I9P101 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I9P101

Prime-Power Transport and a Three-Site Reduction

Payam Danesh, Raoul Bianchetti

Received	Revised	Accepted	Published
11 Jul 2026	17 Aug 2026	10 Sep 2026	28 Sep 2026

Citation :

Payam Danesh, Raoul Bianchetti, "Prime-Power Transport and a Three-Site Reduction," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 9, pp. 1-33, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I9P101

Abstract

The least common multiple of a block of consecutive integers records only the largest prime-power valuation at each prime, whereas the associated product records every valuation. Erdős asked whether two disjoint blocks of the same length can nevertheless have equal least common multiples. This paper develops a prime-power transport method for a hypothetical collision. After removing the universal product-to-LCM defect, the collision has coprime coefficients supported on primes not exceeding the block length. Maximal prime powers of the common LCM then form pairwise coprime transport packets whose residual factors have factorial budgets. A quadratic Taylor lift on diagonal packets produces a primitive positive arithmetic progression in which the reduced gap is forced to square-pack. Maximal-valuation extraction limits all losses to one factorial factor, and a finite irrational square-root spectrum shows that every sufficiently large collision requires at least three active progression sites. Exact certificates eliminate the complete zero-, one-, and two-site regimes for lengths five and six. The method also excludes adjacent blocks and lengths three and four, and yields gap and transport-occupation bounds. The remaining obstruction is a configuration with at least three active sites and a moved transport packet; consequently, the argument gives a rigorous reduction, not a resolution of the conjecture. The bounded length-five search does not determine all rational points and does not exclude solutions beyond its search height.

Keywords

Consecutive Integers, Diophantine Equations, Genus-Two Curves, Least Common Multiple, Prime-Power Transport.

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