Comments on: Checking the ulam spiral http://www.algorithm.co.il/blogs/math/checking-the-ulam-spiral/ Algorithms, for the heck of it Tue, 21 Jun 2011 21:07:08 +0000 hourly 1 http://wordpress.org/?v=3.1.3 By: al deaprix http://www.algorithm.co.il/blogs/math/checking-the-ulam-spiral/#comment-241 al deaprix Thu, 12 Nov 2009 16:47:26 +0000 http://www.algorithm.co.il/blogs/?p=362#comment-241 If you want to see why some diagonals have more primes than others, see my article on www.scribd.com/al%20deaprix. I figured out the modularity of the system about 17-18 years ago, but never got around to doing much with it until recently. The proof that there is differential prime-loading on the diagonals is a very simple one, but one that has been overlooked by those seeking some more complicated answer. If you want to see why some diagonals have more primes than others, see my article on http://www.scribd.com/al%20deaprix. I figured out the modularity of the system about 17-18 years ago, but never got around to doing much with it until recently. The proof that there is differential prime-loading on the diagonals is a very simple one, but one that has been overlooked by those seeking some more complicated answer.

]]> By: A. deAprix http://www.algorithm.co.il/blogs/math/checking-the-ulam-spiral/#comment-240 A. deAprix Mon, 19 Oct 2009 19:06:33 +0000 http://www.algorithm.co.il/blogs/?p=362#comment-240 I am speaking of the progressions of primes. My concern with the spiral square has dealt with primes and prime loading. On the prime loading, I have been able to demonstrate that primes do favor certain diagonals; I did not attempt to find the longest possible progressions. From general scanning of the literature, I have seen it reported that starting with 41 does produce a long progression of primes on one diagonal, but I have not seen a longer one. Finding a longest known progression on a diagonal would be a variant of the problem that seeks to find the longest progression of primes with a given gap; the spiral square, however, has increasing gaps between the elements on a given diagonal, which would complicate the search. I am speaking of the progressions of primes. My concern with the spiral square has dealt with primes and prime loading. On the prime loading, I have been able to demonstrate that primes do favor certain diagonals; I did not attempt to find the longest possible progressions. From general scanning of the literature, I have seen it reported that starting with 41 does produce a long progression of primes on one diagonal, but I have not seen a longer one.

Finding a longest known progression on a diagonal would be a variant of the problem that seeks to find the longest progression of primes with a given gap; the spiral square, however, has increasing gaps between the elements on a given diagonal, which would complicate the search.

]]> By: lorg http://www.algorithm.co.il/blogs/math/checking-the-ulam-spiral/#comment-239 lorg Sun, 18 Oct 2009 06:32:35 +0000 http://www.algorithm.co.il/blogs/?p=362#comment-239 A. deAprix: When you say "no progression will hold" are you referring only to prime numbers, or any kind of progression? I think it's the former, but it would be interesting if you meant otherwise. Also: "one of the longest progressions": can we check that? What number will yield the longest progression? A. deAprix:
When you say “no progression will hold” are you referring only to prime numbers, or any kind of progression?
I think it’s the former, but it would be interesting if you meant otherwise. Also: “one of the longest progressions”: can we check that? What number will yield the longest progression?

]]> By: A. deAprix http://www.algorithm.co.il/blogs/math/checking-the-ulam-spiral/#comment-238 A. deAprix Fri, 16 Oct 2009 19:45:54 +0000 http://www.algorithm.co.il/blogs/?p=362#comment-238 No progression will hold for more than a few elements on a diagonal. Using 41 produces one of the longest progressions. It is very easy to demonstrate that primes do follow a pattern that favors certain diagonals over others. Why that demonstration has eluded mathematicians and interested investigators since 1963 is what is amazing. No progression will hold for more than a few elements on a diagonal. Using 41 produces one of the longest progressions.

It is very easy to demonstrate that primes do follow a pattern that favors certain diagonals over others. Why that demonstration has eluded mathematicians and interested investigators since 1963 is what is amazing.

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