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A pair of twin primes if a pair of primes that are consecutive odd integers.  Find the largest integer that is a divisor of the sum of the two elements in every pair of twin primes, if these two elements are each primes greater than 3.

 Dec 9, 2019
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I don't understand your question! What is the limit on these twin prime? There are infinitely many of them !! Also what do you mean by "  Find the largest integer that is a divisor of the sum of the two elements in every pair of twin primes, if these two elements are each primes greater than 3"?? Do you undestand the question, and if so can you give an example of what you want using 2 small twin primes of 5 and 7 ??

 Dec 9, 2019
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It's a perfectly reasonable question.

The actual number of twin primes is not relevant.

(Has it been proven that they are infinite in number, or is it still conjecture ?)

 

All twin primes are of the form \(\displaystyle 6n\pm 1,\)

(the integer between them has to be divisible by both 2 and 3, so has to be divisible by 6).

Their sum will be of the form 12n, meaning that their sum will be divisible by 12.

 

Could there be a larger common divisor ?

Look at the first few cases.

5   +  7  = 12 = 12*1,

11 + 13 = 24 = 12*2,

17 + 19 = 36 = 12*3.

Clearly, 12 is going to be the largest.

 Dec 10, 2019

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