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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.

Blue
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mathspika
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  • How far have you already checked this conjecture? – Arnaud Mortier Sep 09 '19 at 21:21
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    The usual first step would be to check this pretty far. Keep in mind that there are a lot of small primes, and there are a lot of small twin primes, so small examples don't prove much. – lulu Sep 09 '19 at 21:22
  • up to number 1000 – mathspika Sep 09 '19 at 21:23
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    I would check a lot further than that. As I say, there are a lot of small twin primes. – lulu Sep 09 '19 at 21:26
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    Maybe a bit related to https://math.stackexchange.com/questions/1180982/can-every-even-integer-greater-than-four-be-written-as-a-sum-of-two-twin-primes? –  Sep 09 '19 at 21:26
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    True for all $n\le 104684 = p_{10000}+1$. – rogerl Sep 09 '19 at 21:37
  • speed up in testing if the twin prime used is 5 mod 6, skip 4. –  Sep 09 '19 at 21:43
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    In the linked question, both primes must belong to some twin prime pair, here only one of them. The conjecture here is true upto $n\le 10^7$. – Peter Sep 10 '19 at 08:05
  • Peter, which is the number? – mathspika Sep 10 '19 at 14:30
  • See http://oeis.org/A007534 which provides exceptions to the more strict conjecture and claims no other exceptions $<10^9$. – Steven Clark Sep 10 '19 at 14:39
  • but your link shows for two twin primes in the addends. – mathspika Sep 10 '19 at 20:22
  • @mathspika There is no general rule to find the representation, I just searched an example for ever even number from $6$ to $10^7$. – Peter Sep 11 '19 at 08:03
  • Peter you can do potentially half as much work, see my previous comment. –  Sep 11 '19 at 12:28
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    This is obviously harder than Goldbach. I want to remark that this would also imply that there are infinitely many twin primes. For if $p_1\le p_2\le\cdots\le p_N$ are all the twin primes, you are in trouble when you reach the first prime gap larger than $p_N$. Starting from $(p_N+1)!+2$ at latest a sequence of $p_N$ non-primes follows, and you could not cope.. – Jyrki Lahtonen Sep 15 '19 at 20:10
  • -1 Your conjecture implies both the Goldbach conjecture and the twin prime conjecture. What kind of answer do you expect? – Servaes Apr 05 '20 at 17:48
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    @Servaes a disproof.... – mathworker21 Apr 05 '20 at 18:08
  • Heuristically, a stronger conjecture -- that every sufficiently large even number is a sum of two twin primes -- also seems likely to be true (https://oeis.org/A007534). But as other commenters note, even your weaker conjecture implies Goldbach, so you can't hope for a proof here.... – mjqxxxx Apr 10 '20 at 19:55
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    This question is for mathematics as Terence Tao. – J. Natch Apr 12 '20 at 17:24

2 Answers2

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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.

Roddy MacPhee
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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.

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!

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!

Crystal Wolftron
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