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Old 2008-01-16, 18:47   #1
davar55
 
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Default 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.)
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Old 2008-01-16, 19:31   #2
grandpascorpion
 
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Do the primes have to be consecutive?
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Old 2008-01-16, 20:02   #3
davar55
 
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No, not necessarily consecutive.
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Old 2008-01-16, 20:19   #4
R.D. Silverman
 
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Quote:
Originally Posted by davar55 View Post
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.
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Old 2008-01-16, 20:50   #5
grandpascorpion
 
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Davar55,

You should raise this puzzle with Carlos Rivera over at www.primepuzzles.net. He loves puzzles like this.
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Old 2008-01-17, 01:38   #6
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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
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Old 2008-01-17, 05:02   #7
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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
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Old 2008-01-17, 07:47   #8
lavalamp
 
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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.
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Old 2008-01-17, 13:41   #9
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I checked through 5000. No triply-connected trios found.
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Old 2008-01-18, 01:40   #10
jasong
 
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Too bad I'm not a programmer, sounds like a Linux and Windows version could be relatively easily made, given sufficient skills.
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Old 2008-01-18, 10:52   #11
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Quote:
Originally Posted by davar55 View Post
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.
Is there something especially interesting about this definition?
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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