Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?

Question

Thinking about Goldbach conjecture, I have the following question:

Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?

For example, as 31 and 17 belong to the set of twin primes, 38=31+7 and 40=17+23.

Answer 1

Analyzing this conjecture was really complicated task, but I will pose some constructive arguments and discuss it's consequences.

The proposed conjecture states that Every even number greater than $4$ is the sum of a prime number and another prime number that belongs to the set of twin primes

First of all, it heavily assumes the truth of Goldbach's Conjecture and then the Infinitude of twin primes.

2n = a + b partition form is the mother of two famous conjectures namely The Goldbach's Conjecture and The Twin Prime Conjecture This article will be followed by my humble apologies for not being able to give complete exposition of my above statement, but I'll assure that we won't include it much in our further arguments.

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Consider representing every Positive even Integer $2n$ as sum of $a$ and $b$ where $a,b \in{N}$. For further arguments, we will assume $a\leq{b}$ and $n>2$

Consider $a+b=44$ Define a set $T = \{3,5,7,11,13,17,19\}$ now what if $b$ is non prime for every $a\in{T}$ ?

Obviously the Conjecture will fail. For significantly larger values of $n$, prime numbers are spaced out and if we target the gap length of $2$, they are even more spaced out.

So for very large values of $n$, we could expect that for every $a$ which assumes the value of all twin primes less than $n$ won't get $b$ such that $b$ is any prime and $a+b=2n$

Note that $a$ and $b$ can easily assume all the values of twin primes which are available in the range from $1$ to $\leq{2n-1}$

A close miss Consider $38=a+b$ if we keep $19+19$ aside for a while, we are left with $T= \{3,5,7,11,13,17,19\}$ Except for $7$ all the other values of $a$ have assumed non prime $b$ , so it could be a really really close miss at Infinitude too, nobody knows!

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Aspects of numerical evidences

Let $f(2n)$ denote the total number of ways in which we can represent $2n$ as sum of two primes. Then we have $f(12)=1 , f(18)=2, f(22)=3$ and so on.

We have very compelling numerical evidences for the Goldbach's Conjecture and roughly speaking, the value of $f(2n)$ grows as $n$ grows significantly larger. So it's a good news for all those $a$ which assumes values of twin primes less than $n$

Observe that if there are finite twin primes, then for significantly larger values of $n$ we will fail to get $b$ which is any prime, this is what our arguments and evidences suggests. So the truth of just any one of the Conjecture won't be sufficient to prove the main proposed conjecture.

In Conclusion, there are $3$ essential things which keeps the proposed conjecture alive.

1. Truth of Goldbach's Conjecture and the Twin Prime Conjecture
2. The growth of $f(2n)$
3. $a$ assuming the twin primes, which follows from point $1$ itself.

The most beautiful Aspect is that, in some way, both the Conjectures ( the Goldbach's and the twin prime ) suggests and support our heuristics. But even more interesting would be a Counterexample to it!

Answer 2

Assume the Twin Prime conjecture is True, It's equivalent to there being infinitely many numbers of form $12m$ that have a Goldbach Partition $(6m-1,6m+1)$. Now consider what happens to the values of 3 times the sum of the values of $m$, they get 2 Goldbach partitions for $6m$ equidistant for the higher value. Only when the sums of $m$ values are even, can this continue with 3 times the sum of all $m$ used (with duplication possible) over 2 when it uses 4 indices. So the conjecture, relies on the density of the indices $m$ that sum to even values, being dense enough.