h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Sieve of Eratosthenes - A695991
Author: pikachurinpoche - U189692

I need help with the table. I can't get the spaces the same.

Otherwise?

Thanks.

To make the Table clearer you could give it a border and space it out a bit:



123456
789101112

I hope you don't take your knowledge for granted. Thank you so much. I'll fix it after work, tonite.

I think that there is a problem with this entry.

The method you are using has a very 19th/20th century feel to it.

If that is right, then although it might describe the effect of the sieve very well, it can't be the method that was used.

Greek maths was not very advanced, and this method seems more advanced than they were.

Am I wrong about this?

The Greeks were smart.

Have you seen what they believe about the sieve today? Poor Eratosthenes is either lhao or spinning in his grave.

I'll remove the post to my space.

Still, Pika

Thank you so much. I played with the values and decided on a width of 150. It's great. I can't thank you enough.

Is there a way to write a program to run this, because nobody seems to know about it. When I researched the sites that my web browser found for 'sieve of eratosthenes' none of them know this. I you can run this program, you might find some primes than nobody knows about.

Can a computer program do this? I think it must be able to, because I've seen some programs that follow elaborate patterns.

Anyway, thanks for your help.

Still, Pika

The library of Alexandria. Thats the one that held all the writen knowledge in the world at that time. Just think where'd we be if the Romans Burnt it down. It Boggles the mind. Great entry wish you luck.Time for Coupling ON BBC America That show is Smashing. The KID

I've got nothing to contribute here, but I'm just trying to clear an XML error

Pika,

Your article is not very well explained. I understand what Eratosthenes was doing and the part where you explain that is OK. I can't make out what you are doing with MONAD or that sequence of instructions about moving across and down, ending up on multiples of five. You seem to be using your own terminology, without explaining it. If you have a mathematical idea, you have to explain it in very basic terms so that others can understand.

Whatever you are doing here, if you can say exactly what it is, then it can be programmed in a computer. The only problem is that most computer languages have problems with large numbers. They are designed for business use, so they can rarely cope with numbers of greater than nine digits. Specialised programming tools such as Mathematic can cope with numbers of any size and could be used for investigating large numbers.

I very much doubt you have discovered anything new here. This will not be a good way of discovering new prime numbers, because existing methods are extremely efficient. I've a crude little program I wrote which can test a 20-digit number in less than a day. Try the number 1,111,111,111,111,111,111 for example. It's prime! Your method will run out space long before it reaches 20 hours. Efficient methods can do this check in seconds.

Don't be discouraged, Pika! It's great that you are interested in things like this, but you should not spend too much time on any one topic in Mathematics. Have a look at my "3n+1 Conjecture" in the Guide. Look up Fermat's Little Theorem (not in the guide) and find out about Carmichael numbers. There's plenty to learn.

That should read "long before it reaches 20 digits".

I don't get overly discouraged. I'm not trying to do anything with this except explain it.

I can't cross off diagonals in this programming. It's beautiful in Excel.

If you look at the table, do you see that primes are always in columns headed by the numbers 1 and 5? Make a straight diagonal down through 10, 15, 20, 25, and then around to the other side and the diagonals of 5 continue down and around forever?

Well, if you continue up instead of down from the number 5, you arrive at the square directly above the 6. I didn't know how to build the table with only one box over the 6.

Anyway, can you see that if you begin at that position above the number 6 and go down one square, you will be on the six. Now, from the six, go to the left one square and you will be on the number 5. Now go down one square and you will be in the number 11. Now go left one square and you will be on the number 10. And so on. Can you see the diagonal, proceeding down and to the left of all multiples of 5? You can see that much from the table, right? And you can see that primes will alway be in 2 columns?

You don't need numbers. Just patterns. The numbers can be in base 111,111,111,111,111,111,111 if that's even possible. The sieve doesn't opperate in any one base system. It's the positions of the squares that matter. You can assign them a base 10 value after it pops out as prime. Or you can numerate in any base system you want. It's the positions that have no factors. See?

But I don't know if the computer can do this. Does your method of calculation extract every prime in order, or does it simply check if a given number is prime?

If you take a moment to build a pyramid, you will see what I mean about not needing numbers. I've seen the standard program you are talking about. It's on all the websites.

Thank you for looking.

Still, Pika

Pika,

I can see that the primes are all in those two columns. It's easy enough to prove. The same will apply no matter what width you make the table: certain columns will never have primes, except for once at the very top. The cell you call MONAD is normally called Zero. The multiples of 5 are on the diagonal.

So what? Exactly the same would apply if you made your table any width, for example 12, 13 or 14.

Multiples of 5 are removed on the exact diagaonal going down and left. Likewise on the other side, multiples of base 7 go down and right on the exact diagonal.

But that's not all.

Stay on the right side. The next possible prime down under the 5 in the possible prime's column, is 11. The way you get to 11 is, FROM MONAD, down two possitions, to the 12, and left one position, to the 11. If you continue that pattern, down two positions, you end on 23, left one, 22. And 22 is 2 x 11. And so on, down and around, if you continue this pattern you remove all multiples of 11. The next possible prime under the 11 in the possible primes column is the 17. From MONAD, down 3 left one. That goes on down and around forever, removing all multiples of 17. Similarly, the possible primes in the left hand column, or the ones column, make continuing patterns that factor out multiples of primes in the possible prime columns, thereby leaving only primes untouched.

As I said in the entry, all sizes of grids create patterns. The primes on a grid of base 7 line up on the diagonals. One a sieve of base 12, the multiples of 5 and 7 move like bishops and the multiples of 11 and 13 move like knights, in the game of chess.

However, as the sieve continues downward, the multiples of base 7 shift in sieves higher than base 6. They no longer continue their exact pattern forever.

One reason I'd like to see this programed is to see if it does, in fact, run perfectly forever.

As I said to you before, there is a stability to the number six. Writings said to be from Pythagorus said that the number 6 is perfect. They say it's because 1+2+3 = 1x2x3. I don't think it ends there. There is the repeating nature of the first 60 spaces of a base one sieve.

I have all this in beautiful illustrations on excel, made by my son. I can zip them and send them to you if you want.

[email protected]

This is in the book 'The Secret Teachings of All Ages' by Manly P. Hall; MONAD is like zero, but it is not zero. MONAD is like one but it is not one. MONAD IS GOD.

So, not only is that little square a zero, as is obvious by it's counting position, but it is also a great deal more. Before, I ever started to type the code for my table in the guide entry, that brilliant young man gave me the code. It had a value of 300 for the size of the column. It didn't make a square, and I like squares, so I played around making the number smaller, until the boxes were nearly square.

THAT'S MONAD. MONAD sets the size of the unit. That's how MONAD is like one. Well, how is it like zero?

The zero goes above zero column. It marks the extent of the base line.

In my table, I use base 10 numeration. I didn't numerate in base 6. If I did numerate in base 6, there would be no numeral 6 on the entire table. And, in the far right column, all the numbers will end in 0. This is the 'zero's' column. That's true for every base system. If the grid is base 5 and you numerate in base 5, the far right column will be the zero's column. Not only that, in every case, the rest of the numerals will line up in columns where the last digit is the same. There is a one's column, a two's column, etc.

And notice that MONAD isn't on the table. That's how it's like zero. It's there, lurking invisible, above the far right column, magically showing the end of the base line. The end of the base line is where you go back to the beginning, which in our culture, means returning to the space below the number 1.

When I was trying to explain MONAD to the Dalai Lama's secretary, I said, MONAD is like zero, but it is not zero, MONAD is like one, but it is not one...

At which point, he said, Oh yes, we know if this as well. You speak of the Emptiness. It was then that I realized that the Buddhists are saving Pythagorean information. I know that logic doesn't dictate such faith, but I heard a little voice in my head. It's mostly never wrong.

So when I got back to America, I looked up MONAD in the Math Encyclopedia, and it associated the Yin/Yang with MONAD. I think ADAM/EVE is a MONAD.

I think about this stuff. I care about this stuff. If you didn't know there was a Thirteenth sign of the zodiac, you might think I'm crazy to say there is a little known sign called Ophiucus or Serpentarius. You might not believe me. That doesn't make me wrong. I have many weird findings, some of which seem obvious to me, but I'm not a judge. The sieve is a wonder to behold for me. I was there as I learned about it.

I don't suppose I could get you to obtain a piece of graph paper, four or five aquares per inch, or whatever the measure, as long as there are 60 squares on the long side. Did I explain how to build it well enough? Near the bottom, shade in an entire row across. The next row up, shade every 2nd box. The next row up, shade every third box. The next row every fourth, the next every fifth, and the next every sixth.

The recurring pattern of the first 60 positions will jump right out at you.

If you continue up the pyramid, you will understand primes in a new way. By the time you reach the 20th square up, you are putting only a couple of shaded boxes per row. Only a couple fit on your paper. But this is enough for you to see the sweeping arms, and the lack of factors in the prime columns.

Thank you for your time and energy.

Still, Pika

Pika, Monad is not God, nor is it One. It is Zero. That's where you put in your chart, in the zero position. The reason multiples of 5 make a perfect diagonals is because 5 is one less that 6, the width of your grid. You will get exactly the same pattern no matter what width you use: if your grid is 120 units wide, then multiples of 119 will for a diagonal. There is nothing particularly special about this, it is easy to see and easy to prove.

You don't need to computerise it to see if it goes on forever. Noo matter how far you search, you won't have proved anything. But if you teach yourself some high-school mathematics, you should be able to prove that it goes on forever.

Then why does a sieve on a grid of base 12 experience a shift? The multiples of 7 do not continue in their pattern in the thousands.

Still

Because 7 is not 12 + 1. It is 11 and 13 that form this pattern on a grid with width of 12.

Thank you Pika for explaining the Sieve and for your diagram/graph which I wished I had known about years and years ago. Interesting the discussion you are having with Gnomon - or Gnomon with you. Regards AR1 - I shall get some graph paper!

It's not just the diagonals. It's not just the 5's. There's something stable about base 6 that is not stable in any other base.

Someone else just got graph paper. Did you?

Still, Pika

I used to be surrouned by graph paper.

When I found that a grid of base 7 allowed primes on the diagonal, I built one numbering diagonally, so the primes lined up horizotally.

If you want to see all the charts email me at [email protected] and I can send you a zipped file. If you don't have an unzip, you can get it off the internet. That goes for anyone.

I put the charts together with my theories and derivations. My father ripped it apart with critiques, so if I send it to you, just laugh and look at the charts. My son made the charts. They are excellent.

Thanks!

Still, Pika

Did I get graph paper? No. When you starting naming your squares and claiming they are God, I lost interest. This is not mathematics.

But since you _are_ interested in mathematics, I hope you will persist with the mathematical side. Learn some basic high school mathematics. Then prove that the diagonal pattern for five and seven works. That way you don't have to draw vast diagrams. There is a lot to be learned from looking at patterns in numbers, but unless you can back it up with a proof, it is futile. Pattern spotting is the first step. The next step is to describe it mathematically and the final step is to prove it.