Diagonal prime numbers

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Philosophy Explorer
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Diagonal prime numbers

Post by Philosophy Explorer »

Part of my interest in recreational math are the prime numbers.

Google Ulam's Spiral. It was discovered in 1963 (why not earlier puzzles me).

I discovered a certain number triangle which relates to Ulam's Spiral. The triangle starts off with 2 (it has its good point and weak point). Here's the triangle:

02
03 04 05
06 07 08 09 10
11 12 13 14 15 16 17
18 19 20 21 22 23 24 25 26
27 28 29 30 31 32 33 34 35 36 37
38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65...

Note the ellipsis. Also note that each successive row grows by two numbers. Check the diagonals in this triangle (e.g. 17, 23, 31, 41, 53, 67, 83 and 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227). You will find at least two strings of diagonal prime numbers that relate to quadratic equations just as Ulam's Spiral does. The weakness of the triangle is that its strings aren't as long as the ones in Ulam's Spiral. However it seems to have a higher density of strings compared to Ulam's Spiral.

PhilX
bobevenson
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Re: Diagonal prime numbers

Post by bobevenson »

What do you think of the prime numbers of the Ouzo Cross (http://church-of-ouzo.com/pdf/ouzo-prophecy.pdf)?
Philosophy Explorer
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

Unable to open the PDF (cheap phone).
Can you post up what this cross looks like?

In the meantime what do you think of this triangle?
(ran out of room, otherwise I would have kept going). Nowhere's else on the internet (can that be due to lack of interest?)

PhilX
Ginkgo
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Re: Diagonal prime numbers

Post by Ginkgo »

Philosophy Explorer wrote:Unable to open the PDF (cheap phone).
Can you post up what this cross looks like?

In the meantime what do you think of this triangle?
(ran out of room, otherwise I would have kept going). Nowhere's else on the internet (can that be due to lack of interest?)

PhilX
Prime number theory as Philosophyexplorer presents it requires a theory of all possible prime numbers. If you want to deal with a specific prime number(s) then you would need to put it in the form of an equation.

(edit)
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

Ginkgo wrote:
Philosophy Explorer wrote:Unable to open the PDF (cheap phone).
Can you post up what this cross looks like?

In the meantime what do you think of this triangle?
(ran out of room, otherwise I would have kept going). Nowhere's else on the internet (can that be due to lack of interest?)

PhilX
Prime number theory as Wanderinglands presents it requires a theory of all possible prime numbers. If you want to deal with a specific prime number(s) then you would need to put it in the form of an equation.
I guess you're referring to the quadratic equations such as n-squared + n + 41 (I think Euler came up with this one).

I'm not interested in going in that direction. I would like to see what this Ouzo cross looks like.

Here are some of my thoughts on this. What if Ulam had started his spiral from 2 instead of 1. Would it have improved on the prime number string density?
(in my triangle, I originally started off with 1 and improved on it by starting off with 2 instead). Are there only two prime number strings in my triangle or are there more? Can I improve on the triangle further to at least lengthen the string(s) to completeness?

PhilX
Ginkgo
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Re: Diagonal prime numbers

Post by Ginkgo »

Philosophy Explorer wrote:
I guess you're referring to the quadratic equations such as n-squared + n + 41 (I think Euler came up with this one).


Yes,this was my thinking. I guess 41 would be a good one, but you could start with any prime number.
Philosophy Explorer wrote:
I'm not interested in going in that direction. I would like to see what this Ouzo cross looks like.
As far as I can see it just looks line a horizontal line and a vertical line intersecting.
Philosophy Explorer wrote: Here are some of my thoughts on this. What if Ulam had started his spiral from 2 instead of 1. Would it have improved on the prime number string density?
(in my triangle, I originally started off with 1 and improved on it by starting off with 2 instead). Are there only two prime number strings in my triangle or are there more? Can I improve on the triangle further to at least lengthen the string(s) to completeness?

PhilX
I don't really know, but I can't see how one would ever get horizontal lines and vertical lines intersecting.
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

Ginkgo wrote:
Philosophy Explorer wrote:
I guess you're referring to the quadratic equations such as n-squared + n + 41 (I think Euler came up with this one).


Yes,this was my thinking. I guess 41 would be a good one, but you could start with any prime number.
Philosophy Explorer wrote:
I'm not interested in going in that direction. I would like to see what this Ouzo cross looks like.
As far as I can see it just looks line a horizontal line and a vertical line intersecting.
Philosophy Explorer wrote: Here are some of my thoughts on this. What if Ulam had started his spiral from 2 instead of 1. Would it have improved on the prime number string density?
(in my triangle, I originally started off with 1 and improved on it by starting off with 2 instead). Are there only two prime number strings in my triangle or are there more? Can I improve on the triangle further to at least lengthen the string(s) to completeness?

PhilX
I don't really know, but I can't see how one would ever get horizontal lines and vertical lines intersecting.
Depends on what you mean by intersecting. E.g. would the two lines share the same prime number?

PhilX
Ginkgo
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Re: Diagonal prime numbers

Post by Ginkgo »

Philosophy Explorer wrote:
Ginkgo wrote:
Philosophy Explorer wrote:
I guess you're referring to the quadratic equations such as n-squared + n + 41 (I think Euler came up with this one).


Yes,this was my thinking. I guess 41 would be a good one, but you could start with any prime number.
Philosophy Explorer wrote:
I'm not interested in going in that direction. I would like to see what this Ouzo cross looks like.
As far as I can see it just looks line a horizontal line and a vertical line intersecting.
Philosophy Explorer wrote: Here are some of my thoughts on this. What if Ulam had started his spiral from 2 instead of 1. Would it have improved on the prime number string density?
(in my triangle, I originally started off with 1 and improved on it by starting off with 2 instead). Are there only two prime number strings in my triangle or are there more? Can I improve on the triangle further to at least lengthen the string(s) to completeness?

PhilX
I don't really know, but I can't see how one would ever get horizontal lines and vertical lines intersecting.
Depends on what you mean by intersecting. E.g. would the two lines share the same prime number?

PhilX
Sorry, when I used the word "intersecting" I should have said intersecting at 90degs.
bobevenson
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Re: Diagonal prime numbers

Post by bobevenson »

Philosophy Explorer wrote:Can you post up what this cross looks like?
Just go to http://church-of-ouzo.com, click on to "The Ouzo Prophecy," and then read Notes on the Ouzo Cross.
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

bobevenson wrote:
Philosophy Explorer wrote:Can you post up what this cross looks like?
Just go to http://church-of-ouzo.com, click on to "The Ouzo Prophecy," and then read Notes on the Ouzo Cross.
Still no good Bob. Forces me to download PDFs that I can't open. If someone can post it up here, then I can read it.

PhilX
bobevenson
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Re: Diagonal prime numbers

Post by bobevenson »

Philosophy Explorer wrote:
bobevenson wrote:
Philosophy Explorer wrote:Can you post up what this cross looks like?
Just go to http://church-of-ouzo.com, click on to "The Ouzo Prophecy," and then read Notes on the Ouzo Cross.
Still no good Bob. Forces me to download PDFs that I can't open. If someone can post it up here, then I can read it.

PhilX
Why don't you try your local library.
Philosophy Explorer
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

I'll try another computer tomorrow. Thanks

PhilX
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

I checked the reference given me. The notes merely discuss the cross, but doesn't offer a picture as to what it looks like. I'll keep searching the internet for a picture (btw I don't believe you'll find any religious connections with my triangle).

PhilX
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

I extended the serrated triangle to 290. After checking the diagonals, I discovered this string of diagonal prime numbers:
(197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523). If you check this series, you may see a pattern (along with the other two series). Can you take advantage of the pattern to lengthen the series on either end (beyond the triangle)? Can you associate the series with quadratic equations?

What do you think?

PhilX
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Re: Diagonal prime numbers

Post by Philosophy Explorer »

I've decided to follow up on a question to extend the string of numbers. There's a subtle philosophical point or two to be made. First look at the very short string of 5, 7 and 11 which I didn't list before. One additional prime to add to this string is either 17 or 19, depending on the pattern you use (I prefer the first pattern because that makes it more consistent with the other strings as you will see). So the complete string is 5, 7, 11 and 17. Grouping by patterns is great for math, but not so great for living things.

Now for the other strings I already listed. Let's look at the string of 17, 23, 31, 41, 53, 67 and 83 which has seven numbers in it. Notice that each successive number goes up in this order: 6, 8, 10, 12, 14 and 16. Let's follow this pattern and try to extend the string. So starting from the left-hand side, we subtract 4 from 17, we get 13 which is a prime number and when we subtract 2 from 13, we get another prime number which is 11 and we're done with the left-hand side. With the right-hand side, we add 18 to 83 which equals 101, another prime number, next we add 20 to 101 which equals 121 and here we stop because 121 is composite, not prime and the complete string with prime numbers lying outside of the diagonal is 11, 13, 17, 23, 31, 41, 53, 67, 83 and 101 which has 10 numbers in it. I shall apply the same process to the other two strings to complete them, the completed strings are 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 and 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601.So with the first string I just listed, I added 5 numbers to go from 11 to 16 numbers and with the second string I added 13 numbers to go from 27 to 40 numbers in the completed string (admittedly my serrated triangle stopped at 290 as I had no more space so I had to use the pattern to project the remaining numbers in the last string for that string diagonal and of course I used the diagonal to get the other prime numbers with help of the pattern).

Today I made two more discoveries. With the string of 5, 7 and 11, I was unable to extend it to the left so for that reason and its shortness, I'm not including it with the other strings.

My other discovery is that with every string (including the one of 5, 7 and 11), when extending the string to the right, it extends by just one prime number; when extending the string to the left, the three strings extended by 2, 4 and 12 prime numbers respectively. So I can predict, based on a different pattern, that the next string can be extended one prime number to the right and 48 prime numbers to the left.

PhilX

PS I'm indebted to Beiler for helping out with this thread.
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