Today I delivered a handful of Pilot's Logs to a marina about 9 miles south of the Bear Mountain bridge. Five and a half hours, a little over 150 miles. It was a beautiful day for the drive (although it would have been much nicer to be out of the car), sunny and high 70s. Route 9W along the section just above and below Bear Mountain is very scenic. I pulled into overlooks several times on the way home, including one that overlooks West Point.
I got into an odd train of thought on the Throughway on the way south, concerning infinity. (I've never heard this argument before, so there must be something wrong with it....)
No even numbers (except 2) are prime,
all prime numbers are odd,
odd numbers can therefore be grouped as prime or not prime,
and all non-prime odd numbers are the product of odd numbers.
So all numbers greater than 2 are
- Even, factors are even x even;
- Even, factors are even x odd;
- Odd, factors are odd x odd; or
- Odd and prime
- Therefore there are three times as many even products as there are odd non-prime products.
- Since there are as many odd numbers as even, and only 1/3 of odd numbers are non-prime products, then two thirds of all odd numbers must be prime.
- Therefore the average difference between prime numbers should be less than 4.
- But if you look at a list of the first 1,000 prime numbers, the 1000th prime number is 7919.
- That's an average difference of almost 8.
- There are 105,097,565 prime numbers between 2 and 2,147,483,647 (ref. here).
- That's an average difference of about 20.
- It's reasonable to assume that the difference increases as the numbers go up (primes get more scarce because non-prime multipliers get more common).
- Therefore, there's a huge (impossible) clump of prime numbers up there somewhere taking up the slack,
- or there's a fallacy in the argument - can you spot it?
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