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April 5th, 2010, 11:35 PM   #1
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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
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April 6th, 2010, 05:52 AM   #2
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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.
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April 6th, 2010, 08:24 AM   #3
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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.
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April 6th, 2010, 08:27 AM   #4
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Re: Enchev's conjecture - Goldbach's conjecture for twin pairs

Quote:
Originally Posted by enchev_eg
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).
Why is this surprising?

Quote:
Originally Posted by enchev_eg
This show that may be the twin primes are infinite.
Yes, this conjecture is stronger than the twin prime conjecture. This essentially means there is no hope of proving this conjecture.
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April 6th, 2010, 09:34 AM   #5
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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.
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April 6th, 2010, 10:01 AM   #6
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Re: Enchev's conjecture - Goldbach's conjecture for twin pairs

Quote:
Originally Posted by enchev_eg
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.
Yes, and I think it's neat. I wouldn't have expected this to hold for small numbers, statistically speaking -- but it does. For large numbers it seems all but certain, just like the twin prime conjecture itself.
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April 7th, 2010, 06:15 AM   #7
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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
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April 7th, 2010, 10:47 AM   #8
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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.
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April 7th, 2010, 04:56 PM   #9
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Re: Enchev's conjecture - Goldbach's conjecture for twin pairs

Quote:
Originally Posted by billymac00
well, I would be curious as to any related connection to Highly Composite Numbers? The examples you gave seem to revolve around their multiples.
I doubt it, unless in a trivial sense. Highly composite numbers are sparse, while twin primes are dense. There aren't enough highly-composite numbers for the twins to hang around them.
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April 7th, 2010, 04:59 PM   #10
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Re: Enchev's conjecture - Goldbach's conjecture for twin pairs

Quote:
Originally Posted by enchev_eg
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?
I'm not surprised because I was able to calculate the expected density of these numbers using a model of Cramr. It immediately suggested that there would be only finitely many counterexamples. Some quick calculations with Pari showed that the number of counterexamples was likely to be 0.

Quote:
Originally Posted by enchev_eg
Yes, you are right - proof of this hypothesis will be difficult - but at least something more clearly than "twin primes are infinity"
Well, you might prefer the more specific form of the twin prime conjecture, which (like the Prime Number Theorem) gives the asymptotic number of twin primes up to a given bound.
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