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March 17th, 2013, 08:22 AM  #1 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course  Primitive Pythagorean Triples (PPT)
I forget how, but came across this subject recently, which led me to OEIS A2144. The hypotenuses are primes of the form 4n+1 it says. I then fiddled, making a spreadsheet sequentially thru the 1st 10000 PPT (provided at OEIS) vs the primes as encountered and examining the ratio. Seemingly the ratio seems to be tending towards ~2.15. Perhaps this is well understood or known. However, I then tested several beyond the 10000 by iterating the 2.15 I obtained PPT's at hypotenuses: 225221 factor 2.15022 225241 factor 2.15008 Being too lazy to whip up a Pari script, I then jumped to prime(20000)=224737 and got 483221 factor 2.15016 SO, with this very brief test, the value seems to be holding. I guess my question then would be, if it is a limit, and as the primes are infinite, then the PPTs are infinite essentially? (I know no formally mathematical). Or is this already the case? I ask because I just did a Google search and found stuff like: "...Given a list of 606 primes of the form 4n + 1, might there not be an elegant way to find those two squares? Hmm...." and "...All odd primes are either of the form 4n+1 or 4n+3, and we have seen that both of these arithmetic progressions contain an infinity of primes. However, there is no indication as to the relative growth in numbers of primes for each of these..." What I see seems contrary to these quotes... Thanks [edit: I forgot to credit the site I used which checks the PPTs, http://www.had2know.com/academics/py...0+225221%0D%0A 
March 17th, 2013, 12:45 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Primitive Pythagorean Triples (PPT)
A002144 is infinite, as are the primitive Pythagorean triples. I don't know what ratio you're looking at.

March 17th, 2013, 02:20 PM  #3 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course  Re: Primitive Pythagorean Triples (PPT)
The ratio is effectively the ith hypotenuse/ith prime number. So, continuing from earlier, Prime(200001)=224743. We xpect a triple near hypotenuse 224743*2.15. We find the actual at 483229 (ratio = 2.1501..). Switching to Prime(30000)=350377, the same mechanism yields actual at 753353 (ratio =2.1501..).

March 17th, 2013, 04:42 PM  #4 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Primitive Pythagorean Triples (PPT)
I'm an ardent believer in "empirical mathematics" and I would urge you to keep testing this. It could prove to be an artifact of (relatively) low numbers or it could turn out to be something of real interest. Only time and some mathematical elbow grease will tell. Is there any trend in the deviation from 2.15 that you can determine? 
March 17th, 2013, 05:05 PM  #5  
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course  Re: Primitive Pythagorean Triples (PPT) Quote:
[ CRG had me rev to a newer Pari where I can use the code shown at OEIS ]. I'll try to go higher if time permits  
March 17th, 2013, 05:15 PM  #6 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Primitive Pythagorean Triples (PPT)
Well, my opinion really doesn't matter much, as I am not expert enough in any of the relevant fields. But it certainly LOOKS like you may be on to something quite interesting. Perhaps 2.15 is a rough estimate of what will turn out to be a new irrational constant of theoretical importance like Feigenbaum's constant. From just the examples you gave, it already looks like it might be worth expanding the estimated value to 2.1501 What happens next will be very interesting to find out. Keep us in the loop! 
March 17th, 2013, 05:58 PM  #7 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course  Re: Primitive Pythagorean Triples (PPT)
well, I took it out a bit farther (the first 16384) and it's tending to 2.1419. Sound familiar? Maybe Pi1 ? It may be nothing but always exciting for the briefest of seconds ! 
March 18th, 2013, 05:53 AM  #8  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Primitive Pythagorean Triples (PPT) Quote:
 
March 18th, 2013, 10:10 AM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Primitive Pythagorean Triples (PPT) Quote:
A002144(n)/A000040(n) but 483229 is A002144(20087), not 200001, as far as I can tell. So I must not understand you properly, or else I'm doing my calculations wrong. A002144(n)/A000040(n) tends to exactly 2.  
March 18th, 2013, 03:06 PM  #10 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course  Re: Primitive Pythagorean Triples (PPT)
well, my day job cost me time, and CRG swoops in I had gotton this far: 10000,2.15047 100000,2.11779 200000,2.11015 300000,2.10620 400000,2.10300 500000,2.10244 600000,2.10041 Is this an expected result thinking back on it? Seems interesting to me 

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