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Bi-Dilemma

Do you have a "feeling" for mathematics? That is, given a mathematical statement, do you have a feeling as to its truth value?

For example, I found this "Math Fact" typed at the bottom of a newsletter: "The 100,000,001st prime is 2,038,074,751."

Do you think it is true? Do you have the same feeling about the statement: ""The 100,000,002nd prime is 2,038,094,751."

I admit to having no feeling. But, on the second, I can quickly find the digital sum equal to 39, so I know the statement is false as the number is divisible by 3.

But, all of the divisibility rules for "small" numbers (2-17) fail on the first number. Thus, do I accept it as fact? Could I prove it...yes, and don't want to...

First, I could use brute force and divide by the odd numbers 3 through Sqrt(2,038,074,751) to see if it is prime...even though it would take a long time, with or without technology.

Second, even if the number was prime...how to determine if it is the 100,000,001st prime? This would be a much longer journey.

Finally, after sharing these thoughts, I have decided I do not really care whether it is true or false. Mathematics is about proving or validating more general results. Validating this claim is playing the game of ignoring problem solving and focusing on implementing algorithms far beyond my needs.