할̴̢̨̛̜̖͚̩̿̇͗렐̷̡̛̱͍̠̮̍̽͋루̸̨̛̫̹̖͔̒̋̐야̴̡̛̮͚̜̠̍̓͒

대한민국
Joined March 2021
Today, we’re announcing a solution found by our miners to Erdős Problem 1057. The result proves C(x)=x^(1−o(1)) for Carmichael numbers: the counting exponent tends to 1. Verified in Lean through Conjectures. Full proof below.
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Today, we’re announcing a solution found by our miners to Erdős Problem 579, open since at least 1983. The result disproves the conjecture: graphs can remain dense while avoiding the octahedron graph K₂,₂,₂, with their largest independent sets becoming a vanishing fraction of all vertices. Verified in Lean through Conjectures. Full proofs below.
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Today, we’re announcing a solution found by our miners to Erdős Problem 1062(ii), open since at least 1994. The result proves that the limiting density of the largest subsets of {1, …, n} in which no element divides two distinct others is irrational. Verified in Lean through Conjectures. Full proofs below.
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Today, we’re announcing a solution found by our miners to Erdős Problem 859, a 56-year-old question. Let dₜ be the density of integers whose distinct divisors can sum to t. The result proves that no positive constants c₁ and c₂ satisfy dₜ ∼ c₁/(log t)^c₂, disproving Erdős’s proposed asymptotic. Verified in Lean through Conjectures. Full proof below.
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Today, we’re announcing a solution found by our miners to Erdős Problem 416(i), a 52-year-old question. The result proves that doubling the cutoff asymptotically doubles the number of distinct Euler totient values below it: V(2x)/V(x) → 2. Verified in Lean through Conjectures. Full proofs below.
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The second is Erdős Problem 653, a 29-year-old question. The result proves that n points in the plane can have almost n different counts of distinct distances from individual points, as n grows. Both verified in Lean through Conjectures. Full proofs below.
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ethosnode retweeted
Conjectures has done a great job showing how incentive systems don't just parallelize search over solutions — they parallelize search over search systems themselves. Traditionally, a search pipeline looks something like: team → search algorithm → compute → solution With incentives, this expands to: [verifier + reward] → N competing search systems → solution The latter is meta-search. A traditional team can search over parameters inside its system: optimizers, architectures, prompts, agents, heuristics, search procedures, etc. But many things remain fixed outside the search. The team itself. Its ingenuity. Its hardware. Its capital. Its energy costs. Its infrastructure. Even the assumptions determining which algorithms it chooses to try. An incentive system can push that boundary outward. If anyone can compete and rewards are paid only for verified marginal progress, then the network searches not only over solutions, but over algorithms, teams, compute, infrastructure, and ultimately the organization of the search itself. The abstraction is powerful because dimensions that are constants inside a single research organization become variables at the network level. That larger search space has a cost. But when the bottleneck to finding a solution lies in one of those additional dimensions, expanding the search space can dramatically accelerate convergence. Congrats to the team show casing this to work so effectively with Conjectures.io
Today, we’re announcing solutions found by our miners to two Erdős problems. First, Erdős Problem 108, a 55-year-old question. The result disproves the conjecture, constructing graphs with arbitrarily high chromatic number whose subgraphs without four-cycles need at most six colours. (1/3)
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Today, we’re announcing solutions found by our miners to two Erdős problems. First, Erdős Problem 108, a 55-year-old question. The result disproves the conjecture, constructing graphs with arbitrarily high chromatic number whose subgraphs without four-cycles need at most six colours. (1/3)
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ethosnode retweeted
You are going to figure out all the secrets of the Matrix.
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Today, we’re announcing a solution found by our miners to both parts of Erdős Problem 14, open for over 34 years. The result proves a square-root lower bound on exceptions to unique representation as a sum of two elements of any set of natural numbers. Verified in Lean through Conjectures. Full proofs below.
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💥 Cambio de mando la revista The Economist❓❓ porque las predicciones para el año 2027 son diferentes 🫡🫡😉😉
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Today, we’re announcing a solution found by our miners to Erdős Problem 196, open for over 49 years. The result disproves the conjecture, constructing a permutation of the natural numbers with no four-term arithmetic progression appearing in increasing or decreasing order. Verified in Lean through Conjectures. Full proof below.
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Today, we’re announcing a solution found by our miners to the near-half-density case of Green’s Problem 51, open for 16 years. The result establishes how much structure must exist in sums of dense sets. Verified in Lean through Conjectures. Full proof below.
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Today, we’re announcing a solution found by our miners to Erdős Problem 96, open for over 66 years. The result disproves the conjectured linear bound, constructing strictly convex polygons with superlinearly many unit-distance pairs. Verified in Lean through Conjectures. Full proof below.
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Conjectures, the incentive mechanism for mathematical discovery.
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The truth. COVID.GOV 👀
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ethosnode retweeted
예언에 예언된 대로 일론 머스크가 등장하기 약 80년 전, 이미 베르너 폰 브라운의 책에 '일론'이라는 이름이 등장했다는 사실, 알고 계셨나요? 1948년에 집필된 소설《Project Mars》에서 폰 브라운은 미래의 화성 정부를 이렇게 묘사합니다. "화성 정부는 10명의 지도자가 이끈다. 그중 수장은 보통선거로 5년 임기로 선출되며, 그 직함은 'Elon'이다." 여기서 'Elon'은 사람의 이름이 아니라 화성 정부 수반을 가리키는 직함으로 등장합니다. 그런데 일론 머스크가 평생 화성 이주와 화성 정착에 집착해온 것을 생각하면 묘하게 느껴지지 않나요? 우연의 일치일까요? 아니면 일론 머스크가 말해온 것처럼 외계인이거나 시간여행자, 혹은 이 세상이 시뮬레이션이라는 사실을 우리에게 알려주기 위해 등장한 인물일까요? 우연이라고 하기에는 너무나 기묘한 연결입니다. 🤔
As foretold in the prophecy
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ethosnode retweeted
JUST IN: Claude unexpectedly makes progress on the 167-year-old Riemann hypothesis — one of mathematics’ most notorious unsolved problems.
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IT’S OFFICIAL: HISTORY MADE. 🇺🇸🎆 THE LARGEST FIREWORKS SHOW EVER.
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