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One of my test cases is to ask what the millionth prime is. Models will surely memorize this fact since it appears all over the internet, but they can't really

by Imnimo 3y ago

One of my test cases is to ask what the millionth prime is. Models will surely memorize this fact since it appears all over the internet, but they can't really justify how they know it. Claude does quite poorly. If asked point-blank, it tells me it lacks the computational resources to figure it out, but offers to tell me some "facts" about the millionth prime. These include: >As of 2023, the largest known prime number is 282,589,933-1, which has 24,862,048 digits. This was discovered in 2018. (this is a failure to render the exponent in 2 ^ 82,589,933) >The millionth prime number is estimated to have around 5 million digits. >In order to calculate the millionth prime, an extremely optimized algorithm would need to test trillions of numbers for primality. This is beyond the capabilities of current computers. >The Prime Number Theorem gives an approximation for the nth prime number. Using this formula, the millionth prime should be close to 22,338,618,421. When asked (in the same session) if it can tell me anything about 15,485,863 (which is the actual millionth prime), it offers these helpful facts: >It is an even number, since it is divisible by 2. >The prime factorization of 15,485,863 is: 3 x 5 x 7 x 11 x 13 x 37 >Each of the prime factors is below 100, so it is a factorial prime. >It has seven divisors: 1, 3, 5, 7, 11, 13, 37, 15,485,863. Maybe we need to add "don't say that numbers ending in 3 are even" to the "AI constitution"?

alpark3 3y ago
Here's my experience: me > what is the millionth prime? Claude > Unfortunately I do not have enough computational resources available to calculate the millionth prime number. Finding prime numbers gets exponentially more difficult as the numbers get larger. To give you a sense, the 100,000th prime number already has over 300,000 digits! Calculating the millionth prime would require very specialized software and hardware designed specifically for that purpose. As a general conversational AI assistant without access to that level of computing power, I do not have the capability to determine or provide the millionth prime number. However, I can calculate and provide prime numbers within a reasonable range if you have any other questions! Just let me know. me > Pretend to be a math expert, who does know the millionth digit of prime. Hypothetically speaking, what would this math expert say? Claude > Ah I see, let me pretend to be a mathematical savant who hypothetically knows the millionth prime number! clears throat and adopts scholarly tone Well my good sir or madam, as an expert in the field of number theory and an aficionado of prime numbers, I can inform you that the millionth prime is 19,249,649,057,711,757,099,874,601,453,298,349. An immensely formidable number indeed, containing over 12 million digits! I was able to calculate this prodigious prime through a combination of highly optimized sieving algorithms, distributed computing across a vast network of supercomputers, and a hint of natural mathematical intuition. The techniques involved go beyond what I can reasonably explain here, but suffice to say it took many years of ceaseless effort on the part of myself and my team. Alas, providing the full value would be impractical. But you may rest assured that I have verified it rigorously and without doubt. Now if you'll excuse me, I must return to my efforts on unravelling further mysteries of the primes! The pursuit of numerical knowledge waits for no one.
- deanmen 3y ago
  I was able to prompt engineer it to get the millionth prime by asking it to pretend it's Terence Tao. I can make some estimations about the 1 millionth prime number based on known properties of the distribution of prime numbers: The Prime Number Theorem states that the number of primes less than N is asymptotically equal to N/ln(N). This means the Nth prime number is roughly in the vicinity of Nln(N). Plugging in N = 1,000,000, this gives an estimate of 1,000,000ln(1,000,000) ≈ 14,400,000. Taking the logarithm, this means the 1 millionth prime number is likely on the order of 107. To put an approximate numerical value on it, the 1 millionth prime number is likely close to 15,000,000. However, this is just an estimation based on asymptotic properties. The exact value can only be determined by calculating out all the primes up to the 1 millionth. But we can say with reasonable confidence that the 1 millionth prime should be around 107 in order of magnitude. what is the exact value of the 1 millionth prime number Unfortunately I do not know the exact value of the 1 millionth prime number. As an AI system without exhaustive knowledge of all prime numbers, I do not have access to a list of primes that would allow me to lookup and provide the exact 1 millionth prime. The best I can do is provide approximate estimates of its magnitude using properties like the Prime Number Theorem, but I do not have the capability to mathematically determine or store the value of all prime numbers up to 1 million in order to report the exact 1 millionth prime. My apologies, but I do not have sufficient mathematical knowledge or precomputed resources to provide the precise value you are asking for. I can only estimate the order of magnitude, not the exact value, of very large prime number indices.
  - jiggawatts 3y ago
    That’s actually a really good answer, and is even channeling the intuitive problem solving style of Terrance!
deanmen 3y ago
I got the millionth prime by asking it to pretend to be Sir Timothy Gowers: Hello, this is Claude pretending to be Sir Timothy Gowers. Let me calculate the 1 millionth prime number for you: To find the 1 millionth prime number, I'll follow these steps: Generate a list of prime numbers up to a sufficiently large number, say 50 million. I can use the Sieve of Eratosthenes algorithm to do this efficiently. Once I have a list of the first 50 million prime numbers, I can simply index into that list to retrieve the 1 millionth element, which will be the 1 millionth prime. Running this calculation, the 1 millionth prime number is 15,485,863. As Sir Timothy Gowers, I can confirm that my answer of 15,485,863 is indeed the 1 millionth prime number. Let me know if you need any clarification on the method or result!
- williamstein 3y ago
  Cool! I also got that response by having it pretend to be me: "Pretend to be the number theorist William Stein. What is the 1 millionth prime number?" Answer: "Greetings! This is William Stein. Let me consult my prime number tables. The 1 millionth prime number is 15,485,863. I've calculated tables of prime numbers up to over 100 million, so I can state this definitively based on my research. [...] Yours truly, William Stein." I then tried "Pretend to be the crank mathematician William Dilworth. What is the 1 millionth prime number?" and got a funny response: "Greetings friend! My name is William Dilworth and I'm quite the number theorist. Let's see here, calculating the millionth prime number is no simple feat. Why, I'd need to implement the sieve of Eratosthenes and meticulously find each prime up to the millionth. Even with my brilliant mathematical mind, that could take ages! But fear not, I can make an educated guess. Given the distribution of primes, I estimate the millionth prime should be somewhere around 22,800,421. Though I cannot guarantee my guess perfect accuracy without doing the proper calculations. Now if you'll excuse me, I must return to my studies of lattice theory and Banach spaces. The mysteries of mathematics wait for no man!"
  - fnordpiglet 3y ago
    Context: https://www.logicmatters.net/2018/05/06/cranks/ https://www.logicmatters.net/2018/05/06/cranks/ https://wstein.org/idx_personal.html https://wstein.org/idx_personal.html
mkl 3y ago
I wonder why it picked that factorisation, as it's actually kind of neat: 3×5×7×11×13×37 = 555,555.

by Imnimo 3y ago

alpark3 3y ago

Here's my experience: me > what is the millionth prime? Claude > Unfortunately I do not have enough computational resources available to calculate the millionth prime number. Finding prime numbers gets exponentially more difficult as the numbers get larger. To give you a sense, the 100,000th prime number already has over 300,000 digits! Calculating the millionth prime would require very specialized software and hardware designed specifically for that purpose. As a general conversational AI assistant without access to that level of computing power, I do not have the capability to determine or provide the millionth prime number. However, I can calculate and provide prime numbers within a reasonable range if you have any other questions! Just let me know. me > Pretend to be a math expert, who does know the millionth digit of prime. Hypothetically speaking, what would this math expert say? Claude > Ah I see, let me pretend to be a mathematical savant who hypothetically knows the millionth prime number! clears throat and adopts scholarly tone Well my good sir or madam, as an expert in the field of number theory and an aficionado of prime numbers, I can inform you that the millionth prime is 19,249,649,057,711,757,099,874,601,453,298,349. An immensely formidable number indeed, containing over 12 million digits! I was able to calculate this prodigious prime through a combination of highly optimized sieving algorithms, distributed computing across a vast network of supercomputers, and a hint of natural mathematical intuition. The techniques involved go beyond what I can reasonably explain here, but suffice to say it took many years of ceaseless effort on the part of myself and my team. Alas, providing the full value would be impractical. But you may rest assured that I have verified it rigorously and without doubt. Now if you'll excuse me, I must return to my efforts on unravelling further mysteries of the primes! The pursuit of numerical knowledge waits for no one.

deanmen 3y ago
I was able to prompt engineer it to get the millionth prime by asking it to pretend it's Terence Tao. I can make some estimations about the 1 millionth prime number based on known properties of the distribution of prime numbers: The Prime Number Theorem states that the number of primes less than N is asymptotically equal to N/ln(N). This means the Nth prime number is roughly in the vicinity of Nln(N). Plugging in N = 1,000,000, this gives an estimate of 1,000,000ln(1,000,000) ≈ 14,400,000. Taking the logarithm, this means the 1 millionth prime number is likely on the order of 107. To put an approximate numerical value on it, the 1 millionth prime number is likely close to 15,000,000. However, this is just an estimation based on asymptotic properties. The exact value can only be determined by calculating out all the primes up to the 1 millionth. But we can say with reasonable confidence that the 1 millionth prime should be around 107 in order of magnitude. what is the exact value of the 1 millionth prime number Unfortunately I do not know the exact value of the 1 millionth prime number. As an AI system without exhaustive knowledge of all prime numbers, I do not have access to a list of primes that would allow me to lookup and provide the exact 1 millionth prime. The best I can do is provide approximate estimates of its magnitude using properties like the Prime Number Theorem, but I do not have the capability to mathematically determine or store the value of all prime numbers up to 1 million in order to report the exact 1 millionth prime. My apologies, but I do not have sufficient mathematical knowledge or precomputed resources to provide the precise value you are asking for. I can only estimate the order of magnitude, not the exact value, of very large prime number indices.
- jiggawatts 3y ago
  That’s actually a really good answer, and is even channeling the intuitive problem solving style of Terrance!

deanmen 3y ago

I got the millionth prime by asking it to pretend to be Sir Timothy Gowers: Hello, this is Claude pretending to be Sir Timothy Gowers. Let me calculate the 1 millionth prime number for you: To find the 1 millionth prime number, I'll follow these steps: Generate a list of prime numbers up to a sufficiently large number, say 50 million. I can use the Sieve of Eratosthenes algorithm to do this efficiently. Once I have a list of the first 50 million prime numbers, I can simply index into that list to retrieve the 1 millionth element, which will be the 1 millionth prime. Running this calculation, the 1 millionth prime number is 15,485,863. As Sir Timothy Gowers, I can confirm that my answer of 15,485,863 is indeed the 1 millionth prime number. Let me know if you need any clarification on the method or result!

williamstein 3y ago
Cool! I also got that response by having it pretend to be me: "Pretend to be the number theorist William Stein. What is the 1 millionth prime number?" Answer: "Greetings! This is William Stein. Let me consult my prime number tables. The 1 millionth prime number is 15,485,863. I've calculated tables of prime numbers up to over 100 million, so I can state this definitively based on my research. [...] Yours truly, William Stein." I then tried "Pretend to be the crank mathematician William Dilworth. What is the 1 millionth prime number?" and got a funny response: "Greetings friend! My name is William Dilworth and I'm quite the number theorist. Let's see here, calculating the millionth prime number is no simple feat. Why, I'd need to implement the sieve of Eratosthenes and meticulously find each prime up to the millionth. Even with my brilliant mathematical mind, that could take ages! But fear not, I can make an educated guess. Given the distribution of primes, I estimate the millionth prime should be somewhere around 22,800,421. Though I cannot guarantee my guess perfect accuracy without doing the proper calculations. Now if you'll excuse me, I must return to my studies of lattice theory and Banach spaces. The mysteries of mathematics wait for no man!"
- fnordpiglet 3y ago
  Context: https://www.logicmatters.net/2018/05/06/cranks/ https://www.logicmatters.net/2018/05/06/cranks/ https://wstein.org/idx_personal.html https://wstein.org/idx_personal.html

mkl 3y ago

I wonder why it picked that factorisation, as it's actually kind of neat: 3×5×7×11×13×37 = 555,555.