Problems & Puzzles: Puzzles

Puzzle 1040. Pair of consecutive even integers such that...

Mahdi Meisami sent the following nice puzzle

Due to Goldbach conjecture, It is conjectured that every even N number has some Goldbach representations, for example consider these two consecutive even numbers:

{{7, 23}, {11, 19}, {13, 17}}
{{3, 29}, {13, 19}}
The numbers 13 and 19 are repeated in these representations.

A question came to my mind:

are there consecutive even numbers, N & N+2, without any intersection in their prime representations?.

I examined my conjecture and the following list is its result up to 600:

N={38, 68, 80, 98, 122, 128, 146, 158, 164, 188, 206, 212, 218, 224, 248, 278, 290, 302, 308, 326, 332, 338, 344, 368, 374, 380, 398, 410, 416, 428, 440, 458, 476, 488, 500, 518, 530, 536, 542, 548, 554, 578, 584}
Q1. Does these sequence go to infinity?
Q2. Is there any rule for this sequence? (What about an asymptotic formula?)



During the week May 23-28, 2021, contributions came from: Fred Schneider, Giorgos Kalogeropoulos, Jan van Delden,


Fred wrote:

Let's look at the possible cases mod 30 for a prime + the first of two twin primes. 
There are 8 cases for primes and 3 cases for the first of two twin primes

Modulus 30
First of twin primes
Primes / First of twin primes 11 17 29
1 12 18 0
7 18 24 6
11 22 28 10
13 24 0 12
17 28 4 16
19 0 6 18
23 4 10 22
29 10 16 28

Looking at the combinations and not considering small primes at this step, we find no general modular candidates for overlapping representation for n mod 30  = 2, 8, 14, 20 or 26.

Consider now small primes 3 and 5.  


For n mod 30 = 2, you could have 3  + t and 3 + (t + 2) where  (t, t+2) are  twin primes where t = 29 mod 30 (like you have with n = 32)


For n mod 30 = 14, you could have 3  + t and 3 + (t + 2) where  (t, t+2) are  twin primes where t = 11 mod 30

For n mod 30 = 20,, you could have 3  + t and 3 + (t + 2) where  (t, t+2) are  twin primes where is t = 17 mod 30

For n mod 30 = 26, you could have p + 3 and p + 5 where prime p has mod 30 = 23 


That leaves us with n mod 30 = 8. .  There are no solutions with small prime analysis (except for the trivial first case n = 8 =  3 + 5, n + 2 = 3 + 7)

So, n = 30x + 8 (where x > 0)  and n + 2 never have an overlap in prime representation and thus, there are an infinite number of solutions to the general problem.



Giorgos wrote:

Q2. The answer is yes. There is a rule for this sequence:  
       The numbers N in the sequence are the numbers whose Goldbach representation does not contain any prime that is the lesser of a twin prime (A001359).
       In order to have at least one intersection, one of the primes p_k of the  Goldbach representation of N {{p_1, p_2}, {p_3, p_4}, ... {p_k-1, p_k}} must be a lesser of a twin prime.
       For example if p_3 is a lesser of a twin prime then {p_3 + 2, p_4} will belong to the Goldbach representation of N+2 and p_4 will be the intersection of N and N+2.
       I verified that all of the first 127401 terms of the sequence for N<=10^6 are congruent to 2 mod 6. This is pretty much expected as the Goldbach representation must contain primes of the form
       {prime1, prime2} = {1 mod 6, 1 mod 6} (all the lessers than a twin prime except 3 are congruent to 5 mod 6, so none of them appears in 2 mod 6 representations). 
       I cannot prove that there doesn't exist a N congruent to 0 or 4 mod 6 that belongs in the sequence.
       For example if such a N congruent to 4 mod 6 existed then none of the pairs {prime1, prime2} = {5 mod 6, 5 mod 6} should contain a lesser than a twin prime which is highly improbable as N gets bigger.
       If all the numbers in the sequence are congruent to 2 mod 4 then we should exclude those N for which N-3 is prime. 
       If N-3 is equal to a prime p, then N=p+3 and N+2=p+5 which means that p is the intersection of N and N+2.
       To sum up the rule that I found for this sequence is: 
       All N congruent to 2 mod 6 with N-3 not a prime, belong in this sequence.      

Q1. According to the above answer as N gets bigger and the prime density gets smaller, the probability of N belonging to the sequence gets bigger.
      There exist an infinite number of Ns congruent to 2 mod 6 for which N-3 is not prime. This means that there are infinitely many composite numbers in the sequence {5,11,17,23,29,35,41,47,53,59,65...}
       So, this sequence goes to infinity.   
       Here are my results for the number of terms <N
       Also, I found that a good approximation of the number of terms <N is N/5 - π(Ν) (where π(x) is the prime counting function) but I don't have a proof.     

      N            #N                       N/5 - π(n)
     10^2        4                            -5
     10^3        80                           32
     10^4        1050                       771
     10^5        11860                     10408   
     10^6        127401                   121502
     10^7        1334283                 1335421
     10^8        13785730               14238545                       
     10^9        141241847             149152466


Jan wrote:

A small contribution.




Two even numbers N, N+2, have a common odd prime p in their prime representations if and only if p is the lesser half of a Twin prime.

The numbers N=p+q must therefore have no representation such that either p or q is the lesser half of a Twin prime. It is not necessary to test all representations of N to decide whether or not N qualifies, which does help. In particular if N=p+(p+2), the sum of the two halves of a Twin prime, doesnít qualify.

Finding heuristics for the distribution of the number of non Twin prime splits of N is quite tricky. Although the number of Non Twin primes themselves increase with N, so do the number of representations into two odd primes (which will increase the probability of finding a split where one half is the lesser half of a Twin prime). Or one could try another approach and only focus on a split consisting of a Twin prime p and try to correct for the probability that N-p is prime as well. I thought about it (shortly) and decided to settle for a small experiment.


A graph:


For N<=10^6 (horizontal axis) I counted the number of N that qualify. I called this amount the number of nontwin prime splits.
If N increases we are to expect less and less of these nontwin prime splits per N, i.e. the average probability of finding such a number decreases. Slowly.


A simple fit (on this data) suggests it might have the form: Fraction of Nontwin prime splits/ N = a + b/ln(N)+c/ln^2(N). I found a=6.73, b=-5.40, c=289

I tried this simple model because the density of Twin primes behaves like d/ln^(N) and of primes like 1/ln(N). The fit: R^2=0.994. I didnít perform a sensitivity analysis, but the given values of the parameters are sensitive to the displayed values for small N.





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