Friday, December 14, 2007

How Many Primes Precede------???

One of the things that drive mathematicians crazy is trying to figure out how many prime numbers precede a number. For example if you look up a table of Prime Numbers you will find Prime Number 7919 is the 1000th Prime Number. A prime number is defined as any number that can only be divided evenly by itself and 1. This means that if you suspect a number is a prime then you have to divide it by all the numbers that precede it to see if any number divides into it evenly. The current way of finding the number of Primes preceding any number is by taking the number (7919 for instance) and dividing it by its’ LN or the Log of the Natural Number. The LN or Log of the Natural Number (7919) is (8.977020214). For those of you that don’t have a clue what I’m talking about LN or the Log of the Natural Number (2.718281828) is the number of times that the Natural Number (2.718281828) has to be multiplied by itself to create (7919) which is our present example. Naturally, (2.718281828), doesn’t always multiply by itself evenly to create the number which you want so hence the fraction ( 977020214). I have discovered another way which appears to be consistently closer to the number of primes up to any chosen random number. If you take a look at a list of prime numbers you will discover that, excluding the single digit primes, 1, 2, 3, 5, 7, the far right column of primes end in 1, 3, 7 or 9 once you get to primes of more than one digit. As an example, the following two digit numbers (11, 13, 17, 19) are all primes. Unfortunately, you will soon realize that all numbers ending in 1, 3, 7, 9 aren’t prime numbers. For instance, 21, 33, 27, 39 aren’t prime numbers because each one is divisible evenly by 3. Crazy as it may seem, there is a neat way to get it almost right. I don’t know why it works this way but it seems to be related to chance. You know if you flip a coin it will come up on average over time and many flips as 50% heads or 50% tails. The same principle seems to apply to prime numbers on average. For instance 7919 has 4 digits ( 7, 9, 1, 9 ). Therefore you multiply 7919 by ½ or .5, 3 X. ( 7919 X .5 = 3959.5, 3959.5 X .5 =1979.75, 1979.75 X .5 = 989.875 ) is almost 990 or 10 short of 1000. The secret of how many times to multiply by ½ or .5 is to count the number of digits ( 4 are in 7919 (7, 9, 1, 9 ) and subtract 1 from the total ( 4 - 1 = 3 ). To add to the fun, prime numbers aren’t evenly spaced so for more accuracy choose any arbitrary number ending in 1, 3, 7, 9 in the end column as that arbitrary number may itself be a prime number!!!

If you want to be even more accurate follow these rules for choosing the original number ( 7919 for example )

Prime Numbers have the numerals 1, 3, 7, 9 in their farthest right column which eliminates a lot of numbers that you might think are primes. So choose a number ending in 1, 3, 7, 9.

Not all numbers having 1, 3, 7, 9 in their farthest right column are Prime Numbers, but these numbers can be divided evenly by numbers ending in 1, 3, 7, 9 in the far right column ( 21, 33, 27, 39 ). Try dividing your number by numbers ending in 1, 3, 7, 9 to see if your number is a Prime.

Except for Prime Number 3, the sum of the digits of Prime Numbers never total 3 or multiples of 3. ( 21, 33, 27, 39 ). Add the digits in your chosen number to see of those numbers are divisible by 3. For instance the digits of 69 ( 6 + 9 = 15 ) are divisible evenly by 3 ( 15 / 3 = 5 ). Therefore you know it isn’t a Prime Number.

If you are still curious 7919 / LN (7919) or 7919 / 8.977020214 comes to 882.1412686 or 882 in round figures. 882 is 118 primes short of 1000 which is the number of primes before and including 7919 which is also a prime. My method 7919 X .5 X .5 X.5 = 989.875 or in round figures 990 which is 10 short of the true number of 1,000 or 9 short if you want to think of 999 primes before 7919 which is also a Prime.


Jane said...

Dear Alex,

I'm glad to see you're still hooked on numbers.

Just wanted to wish you a happy holidays and I hope all is going well.

I'm off to catch up on what I've missed out on!


Alex said...

Thanks for wishing me a happy holiday. I see you have graduated and are now moved east. You and I are friends forever and it has all been a great adventure knowing you. See you have shut down your website which is OK because life moves us on to other things. I'll follow your career in movies. I'll watch the newspapers to see how many hearts you've broken and how many young men you have driven to distraction. Anyway you're A-1 in my book!!! Happy Holidays to your toooooo!!!!