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April 5th, 2010, 10:35 PM  #1 
Newbie Joined: Apr 2010 Posts: 13 Thanks: 0  Enchev's conjecture  Goldbach's conjecture for twin pairs Enchev's conjecture If we have one twin prime pair (a1, a2)  the number X = 2 x (a1 + a2) always can be expressed as the sums of two prime numbers X = b1 + c2, that are members of twin prime pairs (b1, b2); (c1, c2). Or another interpretation of Enchev's conjecture: There are always three primes p1, p2, p3 that are members of twin prime pairs (p1, p1+2); (p2, p2 + 2); (p3, p3 + 2)  for which the equation applies p1 = [(p2 + p3 + 2)/ 4]  1 Example: 1. (11, 13); (17, 19); (29, 31) 48 = 2 x (11 + 13) 48 = 17 + 31 = 19 + 29 11 = [(17 + 29 + 2)/4]  1  2. (239, 241); (149, 151); (809, 811) 960 = 2 x (239 + 241) 960 = 149 + 811 = 151 + 809 239 = [(149 + 809 + 2)/4]  1 
April 6th, 2010, 04:52 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 932 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs
It seems to hold (tested up to 100 million). That's not surprising; most numbers should have on the order of x/log^4 x such representations. This gets large rather quickly  there are hundreds of representations for each number at the end of my search.

April 6th, 2010, 07:24 AM  #3 
Newbie Joined: Apr 2010 Posts: 13 Thanks: 0  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs
Things are no so simple. Look at your "hundreds of representations for each number" and notice that EACH twin prime can be represented in relation to two other twin primes, one of which is GREATER than it (first prime). This show that may be the twin primes are infinite.

April 6th, 2010, 07:27 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 932 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs Quote:
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April 6th, 2010, 08:34 AM  #5 
Newbie Joined: Apr 2010 Posts: 13 Thanks: 0  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs
I know this, CRGreathouse, but I'm trying to look for something else in twin primes. I'm tired of looking at the same thing like others, over and over again  and I came up with this  to diversify my day. 
April 6th, 2010, 09:01 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 932 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs Quote:
 
April 7th, 2010, 05:15 AM  #7 
Newbie Joined: Apr 2010 Posts: 13 Thanks: 0  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs
You are clever CRGreathous and understand very well what I want to say. Twin prime conjecture is too FOGGY : TWIN PRIMES ARE INFINITY . OK, this hypothesis is more CLEAR. If for every p1 is the equation applies this mean that ONE of prime number p2 or p3 MUST BE GREATER from p1. It is simple. Yes, you are right  proof of this hypothesis will be difficult  but at least something more clearly than "twin primes are infinity" Never mind! By the way That's not surprising; most numbers should have on the order of x/log^4 x such representations.  Why do you not surprised? What does it mean that you expect such a result statistically for "big" numbers ? EXPECT is not PROOF! If you can proof my conjecture for "big" number, rest easy: "small" numbers we will check with the computers  AND EUREKA!  you will have proof for INFINITY OF TWIN PRIMES! I surprised of everything 
April 7th, 2010, 09:47 AM  #8 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 338 Thanks: 4 Math Focus: primes of course  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs
well, I would be curious as to any related connection to Highly Composite Numbers? The examples you gave seem to revolve around their multiples. 48=4x12, 960=4x240. If so, you might have interest in Potter's work http://primorialconjecture.org/default.aspx, where he ties Twin Primes and HCN. 
April 7th, 2010, 03:56 PM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 932 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs Quote:
 
April 7th, 2010, 03:59 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 932 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Enchev's conjecture  Goldbach's conjecture for twin pairs Quote:
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