Posted: Apr 09, 2010 6:50 am
The following full-post repetition still holds true (EXCEPT FOR TWO THINGS I WILL HAVE TO CORRECT MYSELF), but I'll do a modification to generate the set of primes generated from a given member of the list (rather than just the count, c).
The first thing that needs to be corrected: 'n=n\b' should be 'q=q\b'. So, with that fix, the general case is handled with the above program modified by removing 'print(c)' and replacing 'c+=1' by 'print(m)' (All reference to 'c' is deleted) and the proper choice of n and b.
The outputs for b=4 are blank for all of the numbers listed in my last post except:
n=234099253 (the 4th term) spits out 31331001020311,
n=788894581 (the 20th term) spits out 233001121031311,
n=996511051 (the 25th term) spits out 323121120311023,
n=1417131259 (the 28th term) spits out 1110131323103323,
n=1461773629 (the 30th term) spits out 1113020032110331,
n=1890591799 (the 35th term) spits out 1300230002100313, and
n=2348568403 (the 44th term) spits out 2023333010331103 AND 13030003113231111331113133.
The outputs for the 4th term as b varies on up past b=4 (skipping b=10, as explained) are blank after starting with outputs of 434412134003 and 24104134001242003 for b=5 until b=20 gives 4332833, 172223, 12113, and 2063. I apparently made a mistake thinking there were five primes generated, when in fact there were four (This is the second of the things mentioned at the top). This cuts ever so slightly into the belief that this is a mighty strong coincidence in one way only to increase the nature of the coincidence--around the number 4--in another way. The output for b=21 is blank and b=22 gives 2197633, 98929, 9697, and 2017. b=23 and b=24 give blanks, and b=25 gives 2548003 and 73303. If there is anything at all remarkable going higher up with b, I would be surprised. Even the last part mentioned is barely worth it.
So the features of this coincidence that I discovered are spelled out to the extent I explicitly identified them. I doubt there is a best way to describe it, and I am quite sure mathematicians have not engaged in issues of coincidence internal to mathematics to the degree necessary to analyze it very well. I will be remarking in some way more on this coincidence
some time next week perhaps, but the rest of these posts are going to deal with other mathematical coincidences.
Edit (after five minutes of inhaling legal poison): The sense/belief that I have been mediating the various results is exemplified by the process that brought this coincidence out in full. Following my nose at the Prime Curios website http://primes.utm.edu/curios in order to find something to contribute, I found that thrice prime translation involving base 2 (to base 10, of course) was a rediscovery of mine (If I have it written down, I don't know where, and I would suggest anyone wanting that to just rediscover it also rather than hunting through the site (The number isn't too big)). So, after a few moments thought--realizing that 4 times might not be reasonably quick computing and might likely have been done already--I picked finding twice for bases 2 and 3. In concert with some other thoughts on what might be interesting, I decided on a list rather than just the first. The process of finding the results can be described as having this list come out, checking for the first that got a base-4 result, testing that for base-5, being pleased with that and hoping for more of the same with increasing disappointment until base 9, and then saying to myself that maybe it would continue negatively. I must have been riding pretty high by base 14 or 15, and then a sense of the absurd came over me. When I got to the base 20--and I did not use the program implied here, but one which just output one number to be replaced (explaining my error as probably a typo)--the result I got stunned me. Base 22 doing the same thing (except by the same I mean many, not four specifically) was a further shocker, and I included base 25 for a third more-than-one result. I kept going for quite a ways with no expectations and nothing interesting to see (I don't think). Only later did I see what happened with initial digits and with the positions on the list. I noticed something else I haven't even mentioned as significant first, and that's that the second one to give a base-4 result is all the way up at the twentieth on the list.
Incidentally, I certainly have not had time to investigate deeply the idea that other recreational elementary number theorists have mediated results, but I pulled back from my prediction on my own mediation being stronger than the collective work of others. I'd now say it's more likely that people who have read all of the curios at the above-mentioned website prior to my own and have read the book--the authors themselves, for instance--know more mediated details but couldn't come close to explicitly identifying them as such. I'm really not certain of this though. Where does natural work end and outside assistance begin? I would recommend not looking at anything before 2357 for a hint of what others might have mediated if there is anyone else interested in this, but I might have missed some things earlier on.
Someone wrote:I still don't quite know my PARI/GP well enough to write a program that prompts for an input. Take the first example that I give of what you should write for the second program to test the first number for number of times it converts from base 4 to base ten; and change n accordingly to find the fourth on the list to be the first that generates an output of 1 and the 44th is the first to generate an output of 2, and then go back to the 4th on the list changing b to get all those 0s (remember to skip 10 or you get an endless loop) until at b=20 you get 5, at b=21 another 0, 22-->4, 23-->0, 24-->0, and 25-->2, after which I don't think it was all that interesting. Here is the test generating 0 for the number of times the first on the list generates a prime in going from base 4 to base 10:
{
n=3660109;b=4;c=0;f=1;while(f,q=n;m=0;e=floor(log(q)/log(b));for(i=0,e,d=q%b;n=n\b;m+=d*10^i);if(isprime(m)==1,c+=1;n=m,f=0);next());print(c)
}
The first thing that needs to be corrected: 'n=n\b' should be 'q=q\b'. So, with that fix, the general case is handled with the above program modified by removing 'print(c)' and replacing 'c+=1' by 'print(m)' (All reference to 'c' is deleted) and the proper choice of n and b.
The outputs for b=4 are blank for all of the numbers listed in my last post except:
n=234099253 (the 4th term) spits out 31331001020311,
n=788894581 (the 20th term) spits out 233001121031311,
n=996511051 (the 25th term) spits out 323121120311023,
n=1417131259 (the 28th term) spits out 1110131323103323,
n=1461773629 (the 30th term) spits out 1113020032110331,
n=1890591799 (the 35th term) spits out 1300230002100313, and
n=2348568403 (the 44th term) spits out 2023333010331103 AND 13030003113231111331113133.
The outputs for the 4th term as b varies on up past b=4 (skipping b=10, as explained) are blank after starting with outputs of 434412134003 and 24104134001242003 for b=5 until b=20 gives 4332833, 172223, 12113, and 2063. I apparently made a mistake thinking there were five primes generated, when in fact there were four (This is the second of the things mentioned at the top). This cuts ever so slightly into the belief that this is a mighty strong coincidence in one way only to increase the nature of the coincidence--around the number 4--in another way. The output for b=21 is blank and b=22 gives 2197633, 98929, 9697, and 2017. b=23 and b=24 give blanks, and b=25 gives 2548003 and 73303. If there is anything at all remarkable going higher up with b, I would be surprised. Even the last part mentioned is barely worth it.
So the features of this coincidence that I discovered are spelled out to the extent I explicitly identified them. I doubt there is a best way to describe it, and I am quite sure mathematicians have not engaged in issues of coincidence internal to mathematics to the degree necessary to analyze it very well. I will be remarking in some way more on this coincidence
some time next week perhaps, but the rest of these posts are going to deal with other mathematical coincidences.
Edit (after five minutes of inhaling legal poison): The sense/belief that I have been mediating the various results is exemplified by the process that brought this coincidence out in full. Following my nose at the Prime Curios website http://primes.utm.edu/curios in order to find something to contribute, I found that thrice prime translation involving base 2 (to base 10, of course) was a rediscovery of mine (If I have it written down, I don't know where, and I would suggest anyone wanting that to just rediscover it also rather than hunting through the site (The number isn't too big)). So, after a few moments thought--realizing that 4 times might not be reasonably quick computing and might likely have been done already--I picked finding twice for bases 2 and 3. In concert with some other thoughts on what might be interesting, I decided on a list rather than just the first. The process of finding the results can be described as having this list come out, checking for the first that got a base-4 result, testing that for base-5, being pleased with that and hoping for more of the same with increasing disappointment until base 9, and then saying to myself that maybe it would continue negatively. I must have been riding pretty high by base 14 or 15, and then a sense of the absurd came over me. When I got to the base 20--and I did not use the program implied here, but one which just output one number to be replaced (explaining my error as probably a typo)--the result I got stunned me. Base 22 doing the same thing (except by the same I mean many, not four specifically) was a further shocker, and I included base 25 for a third more-than-one result. I kept going for quite a ways with no expectations and nothing interesting to see (I don't think). Only later did I see what happened with initial digits and with the positions on the list. I noticed something else I haven't even mentioned as significant first, and that's that the second one to give a base-4 result is all the way up at the twentieth on the list.
Incidentally, I certainly have not had time to investigate deeply the idea that other recreational elementary number theorists have mediated results, but I pulled back from my prediction on my own mediation being stronger than the collective work of others. I'd now say it's more likely that people who have read all of the curios at the above-mentioned website prior to my own and have read the book--the authors themselves, for instance--know more mediated details but couldn't come close to explicitly identifying them as such. I'm really not certain of this though. Where does natural work end and outside assistance begin? I would recommend not looking at anything before 2357 for a hint of what others might have mediated if there is anyone else interested in this, but I might have missed some things earlier on.