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The Umans-Wang Arithmetic-Progression Divisor Conjecture

An $n$-divisor set contains a multiple of every integer from 1 to $n$. Umans and Wang proposed, as the arithmetic-progression form of their Strong $(\alpha,\beta)$-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.

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number-theoryAug 7, 2026Significance 18/100Registry: unreviewed

The Umans-Wang Arithmetic-Progression Divisor Conjecture

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An $n$-divisor set contains a multiple of every integer from 1 to $n$. Umans and Wang proposed, as the arithmetic-progression form of their Strong $(\alpha,\beta)$-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.

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An $n$-divisor set contains a multiple of every integer from 1 to $n$. Umans and Wang proposed, as the arithmetic-progression form of their Strong $(\alpha,\beta)$-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.

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