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Tangential is there a list of Unicode characters allowed on hacker news? ﷽!

by singularity2001 3y ago

edf13 3y ago
Yes - apologies for the poor copy-paste here!
- Recursing 3y ago
  You can just ask GPT-4 to rewrite it using unicode characters: Rewrite this post using unicode characters: > The IMO are a bit more involved... Answer: > The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d₁, d₂, ..., dₖ are all the positive divisors of n with 1 = d₁ < d₂ < ... < dₖ = n, then dᵢ divides dᵢ₊₁ + dᵢ₊₂ for every 1 ≤ i ≤ k − 2 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers 𝑛 > 1 that satisfy the given property, we need to closely examine the condition set forth: for a composite number 𝑛 with its divisors d₁, d₂, ..., dₖ where 1 = d₁ < d₂ < ... < dₖ = 𝑛, the property dᵢ divides dᵢ₊₁ + dᵢ₊₂ must hold for every 1 ≤ i ≤ k − 2. Let's break down the problem: 1. Composite Number Requirement: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors. 2. Divisor Property: The property dᵢ | (dᵢ₊₁ + dᵢ₊₂) means that dᵢ₊₁ + dᵢ₊₂ is a multiple of dᵢ. 3. Systematic Examination: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards. Let's start the analysis: - For 𝑛 = 4, the divisors are 1, 2, 4. Here, 1 | (2 + 4), so 4 satisfies the condition. - For 𝑛 = 6, the divisors are 1, 2, 3, 6. Here, 1 | (2 + 3) and 2 | (3 + 6), so 6 satisfies the condition. - For 𝑛 = 8, the divisors are 1, 2, 4, 8. Here, 1 | (2 + 4), but 2 ∤ (4 + 8), so 8 does not satisfy the condition. Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition. The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81. Each of these numbers has the characteristic that for all their positive divisors d₁, d₂, ..., dₖ (where 1 = d₁ < d₂ < ... < dₖ = 𝑛), the condition dᵢ divides dᵢ₊₁ + dᵢ₊₂ is true for every 1 ≤ i ≤ k − 2.
  - CaptainFever 3y ago
    Great idea! I never even thought of using GPT-4 to do this.

Tangential is there a list of Unicode characters allowed on hacker news? ﷽!

edf13 3y ago

Yes - apologies for the poor copy-paste here!

Recursing 3y ago
You can just ask GPT-4 to rewrite it using unicode characters: Rewrite this post using unicode characters: > The IMO are a bit more involved... Answer: > The IMO are a bit more involved... e.g. IMO 2023: > Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d₁, d₂, ..., dₖ are all the positive divisors of n with 1 = d₁ < d₂ < ... < dₖ = n, then dᵢ divides dᵢ₊₁ + dᵢ₊₂ for every 1 ≤ i ≤ k − 2 Although ChatGPT 4 makes a pretty decent start on this already: > To determine all composite integers 𝑛 > 1 that satisfy the given property, we need to closely examine the condition set forth: for a composite number 𝑛 with its divisors d₁, d₂, ..., dₖ where 1 = d₁ < d₂ < ... < dₖ = 𝑛, the property dᵢ divides dᵢ₊₁ + dᵢ₊₂ must hold for every 1 ≤ i ≤ k − 2. Let's break down the problem: 1. Composite Number Requirement: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors. 2. Divisor Property: The property dᵢ | (dᵢ₊₁ + dᵢ₊₂) means that dᵢ₊₁ + dᵢ₊₂ is a multiple of dᵢ. 3. Systematic Examination: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards. Let's start the analysis: - For 𝑛 = 4, the divisors are 1, 2, 4. Here, 1 | (2 + 4), so 4 satisfies the condition. - For 𝑛 = 6, the divisors are 1, 2, 3, 6. Here, 1 | (2 + 3) and 2 | (3 + 6), so 6 satisfies the condition. - For 𝑛 = 8, the divisors are 1, 2, 4, 8. Here, 1 | (2 + 4), but 2 ∤ (4 + 8), so 8 does not satisfy the condition. Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition. The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81. Each of these numbers has the characteristic that for all their positive divisors d₁, d₂, ..., dₖ (where 1 = d₁ < d₂ < ... < dₖ = 𝑛), the condition dᵢ divides dᵢ₊₁ + dᵢ₊₂ is true for every 1 ≤ i ≤ k − 2.
- CaptainFever 3y ago
  Great idea! I never even thought of using GPT-4 to do this.