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The Erdős-Straus conjecture (1948) states that for every integer n ≥ 2 there exist positive integers a, b, c with 4/n = 1/a + 1/b + 1/c. It is empirically verified for n < 10^17 (Salez and others) and closed for many residue classes, but remains open in full generality. We pro...
Google announces the general availability of Imagen 4, its advanced text-to-image model, in the Gemini API and Google AI Studio, featuring significant improvements in text rendering. The new Imagen 4
Andrica's conjecture (1985) asserts An = √p{n+1} − √pn < 1 for every n ≥ 1, where pn denotes the n-th prime. The conjecture is empirically verified for n ≤ 1.3 × 10^16 but remains open. We provide a Mathlib-native Lean 4 formalization with 33 zero-sorry theorems: 24 concrete c...
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Runway Gen-4 delivers sharper visuals, better motion consistency, and more cinematic control than previous models. Read this in-depth 2025 review comparing Gen-4 to Gen-3.
I had the pleasure of speaking with Bill Bengen, creator of the "4% Rule" for retirement planning. Bill has been a reader of Financial Samurai for many years and has always been courteous in the
Commerce Department — The Caribbean Fishery Management Council (Council) submitted Amendment 4 to the Puerto Rico Fishery Management Plan (Puerto Rico FMP) for review, approval, and implementation by NMFS. If approved, Amendment 4 would reclassify rainbow runner from a reef fish to a pelagic fish under the Puerto Rico FMP. Amendment 4 would result in revised management measures for rainbow runner based on the pelagic fish classification, including sector annual catch targets (ACTs) and accountability measures (AMs). The sector-specific annual catch limits (ACLs) would be retained after the reclassification. The purpose of Amendment 4 is to ensure that rainbow runner is managed consistent with its life history characteristics, fishing patterns, and the Council's management of other pelagic species. (59 days remaining)
Environmental Protection Agency — The U.S. Environmental Protection Agency (EPA) is reconsidering the Tier 4 criteria pollutant standards for new motor vehicles promulgated within the final rule entitled "Multi-Pollutant Emissions Standards for Model Years 2027 and Later Light-Duty and Medium-Duty Vehicles." This reconsideration will occur in two separate, but coordinated, rulemakings. In this Part 1 rulemaking, the EPA is proposing to amend the phase-in schedule for the Tier 4 criteria pollutant standards for certain vehicles to address changing circumstances and feasibility concerns. These amendments, if finalized, would extend the Tier 3 standards for certain vehicles to model years (MYs) 2027 and 2028 such that the Tier 4 standards for these vehicles would phase in starting with MY 2029. The EPA is also proposing other changes to the test protocols used to evaluate emissions performance for certification and related regulatory issues. Potential amendments to the Tier 4 standards and other program elements will be proposed separately in a future Part 2 rulemaking. (48 days remaining)
