The likelihood that the product of two different numbers is a prime number is essentially zero. This is because a prime number is defined as a number that has exactly two distinct positive divisors: 1 and itself¹². Therefore, the product of two different numbers (both greater than 1) will always have at least four distinct positive divisors: 1, each of the numbers, and the product itself. This means it cannot be a prime number.
As for how this changes as the numbers get larger, it doesn't. Regardless of how large the two different numbers are, their product will never be a prime number if both numbers are greater than 1. This is a fundamental property of prime numbers and does not change based on the size of the numbers³.
However, if one of the numbers is 1, then the product will be a prime number if the other number is a prime number. But in this case, the two numbers are not different if we consider 1 not to be a prime number, which is the standard definition².
In terms of probability, the probability that two different integers are both simultaneously divisible by a prime p is $$\frac{1}{p^2}$$⁵. This means that the probability that two different integers are not simultaneously divisible by a prime p is $$1 - \frac{1}{p^2}$$⁵. This might give some insight into the distribution of prime numbers, but it doesn't directly answer your question because it's about divisibility, not about the product being prime.
I hope this helps! Let me know if you have any other questions..
Source: Conversation with Bing, 12/12/2023
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