Bounds for the 3x+1 problem using difference inequalities
Ilia Krasikov, Jeffrey C. LagariasActa ArithmeticaPublished 2003doi 10.4064/aa109-3-4Original︎ ↗Archived copy
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- For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.
Supports. The primary source. The paper establishes that feasibility of a linear program associated to Krasikov's difference inequalities yields exponential lower bounds for their positive monotone solutions, then reports a computation for the modulus 3^11 system giving the exponent 0.84; the bound holds for all sufficiently large x, and more generally for integers whose orbit contains any fixed value not divisible by 3. The result was refereed and published in Acta Arithmetica in 2003. Sep 17, 2026
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Abstract: We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities. We give an improvement which apparantly extracts full information from the inequalities. By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.
Supports. The primary source. The paper establishes that feasibility of a linear program associated to Krasikov's difference inequalities yields exponential lower bounds for their positive monotone solutions, then reports a computation for the modulus 3^11 system giving the exponent 0.84; the bound holds for all sufficiently large x, and more generally for integers whose orbit contains any fixed value not divisible by 3. The result was refereed and published in Acta Arithmetica in 2003.
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Record
- Kind
- journal article
- Authors
- Ilia Krasikov, Jeffrey C. Lagarias
- Publisher
- Acta Arithmetica
- Published
- 2003
- DOI
- 10.4064/aa109-3-4
- Link
- https://arxiv.org/abs/math/0205002
- Archived copy
- https://web.archive.org/web/20260817224212/https://arxiv.org/abs/math/0205002
- Stored text
- 4,085 characters, retrieved Sep 17, 2026 · 47 sections, 55 passages
- Claims extracted
- Not recorded
- Versions
- One known
- Copies
- None recorded
- Corrections
- None recorded
- Facts checked
- Nov 9, 2026
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