Left Hand Of The Electron by Asimov Isaac

Author:Asimov, Isaac [Asimov, Isaac]
Language: eng
Format: epub
Published: 0101-01-01T00:00:00+00:00

PRIME QUALITY

We could then grind out an infinite series of known primes by turning a formula-crank, even though we know we are skipping an infinite quantity of other primes.

Again, no (except for some specialized non-useful cases). No matter how we try to find a useful system that will yield primes only, non-primes will always sneak in. For instance, you might think that adding 1 to the product of successive primes beginning with 2 might yield only prime numbers. The numbers I got this way a little earlier in the article were 7, 31, 211, 2311—all primes! But then, the next in the series was 30,031, and that was not a prime.

Formulas have been worked out in which the value n was substituted by the numbers 1, 2, 3, and so on, with prime values obtained for every value up to n = 40. And then for n = 41, a non-prime will pop out.

So let’s lower our sights again. Is there any system that will allow us to crank out only non-primes? Non-primes may not be interesting but at least we can get rid of them and study a group of remaining numbers that will be denser in primes.

Yes! At last we have something to which the answer is, yes! In working out the sieve of Eratosthenes, for instance, perhaps you noticed that in crossing out every second number after 2, you crossed out only numbers that ended with 2, 4, 6, 8, and o, and that you crossed out every number that ended with 2, 4, 6, 8, 0. This means that any number, no matter how long and formidable, even if it has a trillion digits, is not a prime if the last digit is 2, 4, 6, 8, or o; if it’s an “even number,” in other words.

Since exactly half of all the numbers in any finite successive list end in these digits, that means that all primes (except for 2 itself, of course) must exist in the other half—the odd numbers.

Then again, when you begin with 5 and cross out every fifth number, you cross out only numbers that end with 5 and o, and every number that ends with 5 and 0. Numbers ending with o are already taken care of, but now we can eliminate any number from the list of possible primes if the last digit is 5 (except for 5 itself, of course).

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.