Claude 2
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&#x27;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 &quot;facts&quot; about the millionth prime. These include:<p>&gt;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)<p>&gt;The millionth prime number is estimated to have around 5 million digits.<p>&gt;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.<p>&gt;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.<p>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:<p>&gt;It is an even number, since it is divisible by 2.<p>&gt;The prime factorization of 15,485,863 is: 3 x 5 x 7 x 11 x 13 x 37<p>&gt;Each of the prime factors is below 100, so it is a factorial prime.<p>&gt;It has seven divisors: 1, 3, 5, 7, 11, 13, 37, 15,485,863.<p>Maybe we need to add &quot;don&#x27;t say that numbers ending in 3 are even&quot; to the &quot;AI constitution&quot;?