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 2008-01-16, 18:47 #1 davar55     May 2004 New York City 2×2,099 Posts Connected Primes Call a trio {a,b,c} of primes singly-connected if all three of {a',b',c'} are prime, where a' = 2a+bc, b' = 2b+ac, c' = 2c+ab. For example, {3,5,7} are singly-connected, because {41,31,29} are prime. Call the trio doubly-connected if {a',b',c'} are singly-connected, and triply-connected if {a',b',c'} are doubly connected. The puzzle is: find a triply-connected trio {a,b,c} of primes. (I present this as an open question because I haven't yet.)
 2008-01-16, 19:31 #2 grandpascorpion     Jan 2005 Transdniestr 503 Posts Do the primes have to be consecutive?
 2008-01-16, 20:02 #3 davar55     May 2004 New York City 2×2,099 Posts No, not necessarily consecutive.
2008-01-16, 20:19   #4
R.D. Silverman

Nov 2003

26×113 Posts

Quote:
 Originally Posted by davar55 Call a trio {a,b,c} of primes singly-connected if all three of {a',b',c'} are prime, where a' = 2a+bc, b' = 2b+ac, c' = 2c+ab. For example, {3,5,7} are singly-connected, because {41,31,29} are prime. Call the trio doubly-connected if {a',b',c'} are singly-connected, and triply-connected if {a',b',c'} are doubly connected. The puzzle is: find a triply-connected trio {a,b,c} of primes. (I present this as an open question because I haven't yet.)
By Schinzel's Conjecture (A Generalization of the K-tuples conjecture),
there should be ininfinitely many such sets. However, they will be
very sparse.

You want a,b,c, 2a+bc, 2b+ac, 2c+ab, 2(2a + bc) + (2b+ac)(2c+ab),
2(2b+ac) + (2a+bc)(2c+ab), 2(2c+ab) + (2a + bc)(2b+ac), etc.
to all be prime.

This a lot of conditions on simultaneous primality. If a,b,c are near
N, then a', b', c' are near N^2 and a'', b'', c" are near N^4, so the
probability of all being prime is
P ~ 1/log^3 N * 1/log^3 (N^2) * 1/log^3(N^4)
and this is quite small. You want the number of such sets less than N to
be at least 1, so we require N*P > 1, on average to find such a set. N
will have to be quite big.

 2008-01-16, 20:50 #5 grandpascorpion     Jan 2005 Transdniestr 1111101112 Posts Davar55, You should raise this puzzle with Carlos Rivera over at www.primepuzzles.net. He loves puzzles like this.
 2008-01-17, 01:38 #6 lavalamp     Oct 2007 London, UK 22×3×109 Posts Well I've written a 100 line piece of code in JavaScript (it's the language I'm most fluent in) and determined that when all three arguements are less than 1000, there are 736 singly connected triplets, but no doubly or triply connected triplets. Of those 736, this if the largest one: 3, 991, 997 988033, 4973, 4967 I may translate my code into C which will allow me to check bigger numbers much more quickly. An interesting thing I noticed is that the overwhelming majority of the results (695 out of 736) have 3 as the first arguement, with the remainder (41 out of 736) having 5 as the first. Edit: On a whim I just checked arguements up to 10000 and there are still no doubly connected triplets. Last fiddled with by lavalamp on 2008-01-17 at 01:56
 2008-01-17, 05:02 #7 grandpascorpion     Jan 2005 Transdniestr 503 Posts I think there's a problem with your code. 7, 11, 13 is singly-connected. It gives you : 157, 113, 103, all prime I found a bunch of doubly-connected trios: Examples: 5, 7, 709 7, 11, 3967 Looking for triply-connected trios now ... P.S. You should try PARI. It's great for number crunching like this. Last fiddled with by grandpascorpion on 2008-01-17 at 05:10
 2008-01-17, 07:47 #8 lavalamp     Oct 2007 London, UK 22·3·109 Posts Ah, I made a slight error, breaking a loop where I shouldn't have been breaking a loop, all fixed now. The abundance of 3's with a few 5's for the first arguement was a symptom of the problem. I now find 9815 singly connected and 68 doubly connected triplets when the size limit for the arguements is 1000. I also find 39529 singly and 206 doubly triplets up to 2000. I've not checked for triply triplets because the numbers are simply too big. Here are the last two singles to be outputted: Code: 257, 1993, 1999 3984521, 517729, 516199 613, 1997, 1999 3993229, 1229381, 1228159 And the last two doubles: Code: 521, 1553, 1997 3102383, 1043543, 813107 848518322867, 2522571421067, 3237471689183 19, 1259, 1999 2516779, 40499, 27919 1135725139, 70266033899, 101927088559 I had a read of the PARI article on wikipedia, it looks very interesting, I think I'll have a play around with that.
 2008-01-17, 13:41 #9 grandpascorpion     Jan 2005 Transdniestr 503 Posts I checked through 5000. No triply-connected trios found.
 2008-01-18, 01:40 #10 jasong     "Jason Goatcher" Mar 2005 5×701 Posts Too bad I'm not a programmer, sounds like a Linux and Windows version could be relatively easily made, given sufficient skills.
2008-01-18, 10:52   #11
davieddy

"Lucan"
Dec 2006
England

144638 Posts

Quote:
 Originally Posted by davar55 Call a trio {a,b,c} of primes singly-connected if all three of {a',b',c'} are prime, where a' = 2a+bc, b' = 2b+ac, c' = 2c+ab.
I note that a,b,c must all be different and odd.

I think I follow RDS's post for a change:

There are ~N^3/6 triples with a limit of N on each member.
The probability of them all being prime is ~(1/ln(N))^3.

a',b',c' are ~N^2 but given a,b,c are odd primes, a',b',c' must
be odd. So the probability all of them being prime is ~(2/ln(N^2))^3
=(1/ln(N))^3.

So the expected number of singly connected triples is

N^3/(6(ln(N))^6)

Does this tally with the data tested so far?

David

Last fiddled with by davieddy on 2008-01-18 at 11:20

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