Problems & Puzzles: Puzzles

Puzzle 1075 A conjecture about twin primes Long time ago, Dmitry Kamenetsky sent the following nice puzzle, that I forgot in my files: Conjecture: For every average of twin primes M there exists a twin prime p<M such that pM is another average of twin primes. Examples: M=4: 4 is an average of twin primes (3, 5), and 4*3=12, which is also an average of twin primes (11,13). 3 is a twin prime less than 4. M=6: 6*3=18, 6*5=30. M=12: 12*5=60 M=18: 18*5=90, 18*11=198 Q. Can you prove this conjecture or find a counterexample?

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During the week 13-19 Feb 2022, contributions came from Ivan Ianakiev, Adam Stinchcombe, Paul Cleary, Emmanuel Vantieghem, Fausto Morales.

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Ivan wrote:

My unremarkable  remark is that to ask for a proof of this conjecture is to ask for a proof of Twin prime conjecture (that there are infinitely many prime pairs).

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Adam wrote:

If you can resolve the conjecture in the affirmative, then you can prove the twin prime conjecture.  Let M be an average of two twin primes, and find the member p of the twin primes such that pM is also an average of two twin primes.  By the conjecture pM is the average of two twin primes, so there is a member p’ of the twin primes such that p’(pM) is the average of two (bigger than the previous so new) twin primes, then p”(p’M) is the average of two new twin primes, etc., off to infinity.

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Paul wrote:

M = 108 fails the test. 

 

From twin primes 107 and 109.  All twin primes p less than 108 are {3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107}, giving the following pM list of {324,540,756,1188,1404,1836,2052,3132,3348,4428,4644,6372,6588,7668,7884,10908,11124,11556}, non of them being the average of another twin prime.

I couldn’t find any other counterexamples up to and including the twin primes {22668671, 22668673} or an M value of 22668672.  There were 239532 pM values at this point.

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Emmanuel wrote:

The conjecture is not true for  M = 108
The set of twin primes < 108  is : 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107.
Multiplied with  108 gives : 324,540,756,1188,1404,1836,2052,3132,3348,4428,4644,6372,6588,7668,7884,10908,11124,11556.
None of these numbers is the center of a twin prime pair.
But, I think that's the only exception. 

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Fausto wrote:

This conjecture is true for all M with 108 < M < 10^13.

In addition, the following chain of 11 consecutive cases (with M > 10^390), starting at

(p, M) = (43663031, (2^541) * (3^476))

and up to

(p, M) = (177350819, (2^541) * (3^476) * 43663031 * 42850021 * 43223671 * 63207499 * 63352411 * 65585911 * 65949029 * 81744979 * 97245989 * 107005051)

appears to indicate that it is probably safe to place far stronger demands on the prime p required -
 

For example:

 

p < 10^{[log10 (M^2)]^(1/3)} ,
 

also true for all 108 < M < 10^13.

 

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