A Prime Number is a number that is defined as only being divisible by itself and 1. On the surface it would seem that this simple definition should produce a simple, predictable pattern for Prime Numbers but it doesn’t happen. It doesn’t happen because this simple definition is subject to about 4 rules that complicate the location of a Prime Number.

Here are the rules:

1. Numbers that are evenly divisible by 2 aren’t Prime Numbers. These numbers end in 0, 2, 4, 6, 8 in the far right column.

2. Numbers that are evenly divisible by 5 aren’t Prime Numbers. These numbers end in 5 and 0 in the far right column. (25, 40)

3. Numbers ending in 1, 3, 7, 9 may or may not be Prime Numbers. (11, 13, 17, 19) are Prime Numbers since they are only divisible by themselves and 1. (21, 33, 27, 99) are not Prime Numbers because they are divisible by some number having 1, 3, 7, 9 in the far right column

4. Except for the single digit Prime Number 3, any number whose digits total 3 or a multiple of 3 and has 1, 3, 7, 9 in the far right column isn’t a Prime Number ( 69, 6 + 9 = 15, 15 / 3 = 5 ) isn’t a Prime Number

If only the first two rules existed then Prime Numbers always ending in 1, 3, 7, 9 would be evenly distributed. Unfortunately, some numbers ending in 1, 3, 7, 9, in the far right column can be evenly divisible by some other numbers ending in 1, 3, 7, 9. In addition, except for the single digit Prime Number 3, any number having 1, 3, 7, 9 in the far right column and whose digits total 3 or a multiple of 3 isn’t a Prime Number. The 3rd rule is bad enough but the 4th rule really scrambles the Prime Number Pattern.

## Wednesday, December 26, 2007

## 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.

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.

## Friday, December 07, 2007

### Weird Spread

One of the things that drive people nuts that enjoy puzzles are Primes. Primes are any number that can be divided evenly by only itself and 1. If you ever look at a Table Of Primes, you will see they seem to have a pattern but there isn’t any readily recognizable rhyme or reason to it. Logically, you would think that the spaces that would separate Primes would be even or at the most not far off even. Here’s what I mean. The simplest Primes are the single digits 1, 2, 3, 5, 7. Looks relatively simple to me, but the larger the numbers, the more the Primes’ locations seem to be completely disorganized. Since Primes are defined as numbers divisible by themselves and 1 you can eliminate some numbers immediately. Two (2) divides evenly into any number that has 0, 2, 4, 6, 8 in the far right column ( 10, 12, 24, 36, 48, for example ). Five (5) is the only other number that divides evenly into any number that has 5 or 0 in the far right column (125, 155, 175 ), ( 120, 150, 180). The remaining prime numbers are 1, 3, 7. They may or may not divide evenly into any number that has a 1, 3, 7 in the far right column. The only other number not discussed is 9 which isn’t a prime number because it’s divisible by 3 ( 9 / 3 = 3 ) Weird as it is, you will find that the numbers 1, 3, 7, and 9 are in the farthest right column of Prime Numbers. For example, ( 11, 13, 17, 19 ) are all Prime Numbers. If you divide any number ending in 1, 3, 7, 9 in the far right column into any number ending in ( 1, 3, 7, 9 ) in the far right column sometimes it divides evenly which means that particular number that ends in ( 1, 3, 7, 9 ) isn’t a Prime Number, For example, ( 21, 33, 27, 39 ) are all divisible by 3. You will see that the Prime Number locations would now start to spread out because not all numbers ending in ( 1, 3, 7, 9 ) in the far right column are only divisible by themselves and 1. The final randomizer is due to the fact that, except for single digit prime number 3, the total of the digits forming a Prime number ending in 1, 3, 7, 9, will never total 3 or a multiple of 3. For instance 69 is not a Prime because its’ digits 6 and 9 total 15 which is divisible evenly by 3. ( 69, 6 + 9 = 15, 15 / 3 = 5 ).

So, in summary, randomizing occurs because:

Prime Numbers have 1, 3, 7, 9 in their farthest right column leaving spaces of 1, 2, multiples of 2 or even odd or even powers of 2 between Prime Numbers.

Not all numbers having 1, 3, 7, 9 in their farthest right column are Prime Numbers ( 21, 33, 27, 39 ).

Except for Prime Number 3, the sum of the digits of Prime Numbers never total 3 or multiples of 3. ( 21, 33, 27, 39 )

So, in summary, randomizing occurs because:

Prime Numbers have 1, 3, 7, 9 in their farthest right column leaving spaces of 1, 2, multiples of 2 or even odd or even powers of 2 between Prime Numbers.

Not all numbers having 1, 3, 7, 9 in their farthest right column are Prime Numbers ( 21, 33, 27, 39 ).

Except for Prime Number 3, the sum of the digits of Prime Numbers never total 3 or multiples of 3. ( 21, 33, 27, 39 )

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