Questions tagged [goldbachs-conjecture]

For questions about Goldbach's conjecture: every even integer greater than two is the sum of two primes.

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parity problems for sieve methods, is it only for Selberg Sieve or for all sieve methods?

It is said that sieve methods have parity problems. Terence Tao gave this "rough" statement of the problem: "Parity problem. If A is a set whose elements are all products of an odd number of primes ...
david's user avatar
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17 votes
9 answers
1k views

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
david's user avatar
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15 votes
3 answers
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Evidence against Goldbach's Conjecture?

It recently occurred to me that, unless I'm much mistaken, Goldbach's conjecture can easily be seen to be equivalent to a seemingly more general statement: Every number $n$ divisible by any $1<d&...
Damian Reding's user avatar
5 votes
1 answer
1k views

Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
Lehs's user avatar
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4 votes
1 answer
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Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?

I am trying to find a counterexample for the following expression when $d=6$. ($G(n)$ = Goldbach partition of the even number $n$) ${\forall}$ n=2*k / k${\in}$N, n${\geq}$8 ${\exists}$(${p_i}$,${p_j}...
iadvd's user avatar
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2 votes
5 answers
1k views

Is there any known relationship between Goldbach's comet G(n) and the prime counting function (${\pi(n)}$)?

The "extended" Goldbach conjecture defines R(n) as the number of representations of an even number n as the sum of two primes, but the approach is not related directly with ${\pi(n)}$, is there any ...
iadvd's user avatar
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2 votes
2 answers
258 views

Can sieve method prove ternary (three) prime Goldbach conjecture?

Can sieve method prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem) ? I had done some research, I could not find any articles on this. Can anyone provide some help on this ? I ...
david's user avatar
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4 votes
0 answers
258 views

Any heuristic explanation on why sieve methods can not prove Goldbach conjecture?

Any heuristic explanation on why sieve methods can not prove strong Goldbach conjecture ? I remember that Terence Tao published a blog and gave an heuristic explanation on why circle methods very ...
jim's user avatar
  • 49
3 votes
0 answers
321 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
jim's user avatar
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3 votes
2 answers
653 views

Any results for small number Goldbach conjecture research?

It seems to me that most research results on Goldbach conjecture research are for large number. (Example: results of Vinogradov, Terence Tao, Harald Helfgott, etc). My understanding is that those ...
jim's user avatar
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5 votes
2 answers
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What does it mean for a theorem to be "almost surely true", in a probabilistic sense? (Note: Not referring to "the probabilistic method")

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them ...
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19 votes
1 answer
3k views

Books to read to understand Terence Tao's Analytic Number Theory Papers

I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ...
jim's user avatar
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1 vote
1 answer
80 views

Any books on Trigonometrical Sums (for the Theory of Numbers )?

All: Can anyone recommend good books on Trigonometrical Sums ? The only book I found is Vinogradov's book: Method of Trigonometrical Sums in the Theory of Numbers. but it is really old. I am ...
ben's user avatar
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1 vote
0 answers
186 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
ben's user avatar
  • 27
-1 votes
2 answers
561 views

Goldbach conjecture - what's wrong with this "proof"?

Can somebody please tell me what is wrong with the following "proof" of the strong Goldbach Conjecture? For every even number $n$, there are $\frac{n}{4}$ pairs of odd numbers $[a, b]$ such that $a+b=...
tomerius's user avatar
7 votes
1 answer
595 views

What are Goldbach conjecture for other algebra structures, matrix, polynomial, algebraic number, etc?

All: what are Goldbach conjectures for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, and other algebraic structures ? In other word, for other algebraic ...
ben's user avatar
  • 71
4 votes
1 answer
330 views

What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
ben's user avatar
  • 331
5 votes
2 answers
1k views

How many even numbers are the sum of at most one pair of prime numbers?

Consider the set of all even numbers larger than $2$. Goldbach's conjecture states that every element is the sum of a pair of prime numbers. It has not been proved that all elements abide to that ...
barak manos's user avatar
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6 votes
2 answers
232 views

Is this Goldbach-type problem easy to solve?

Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ? This comment informs that it's an obvious ...
Sebastien Palcoux's user avatar
26 votes
1 answer
4k views

Could it be that Goldbach conjecture is undecidable?

The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach ...
Yes's user avatar
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0 votes
1 answer
248 views

Goldbach's Weak Conjecture

I have a few questions on GWC, as the Wikipedia's page on it appears to be somewhat incomplete. Which of the following two statements is considered as the actual GWC? Every odd number greater than 5 ...
barak manos's user avatar
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4 votes
1 answer
179 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
Mayank Pandey's user avatar
0 votes
1 answer
358 views

Questions about the proof that every odd integer is the sum of 5 primes

In http://arxiv.org/pdf/1201.6656.pdf, Tao proved that all odd numbers greater than 1 are the sum of 1, 3, or 5 primes. In page 36-37, he uses the fact that for all $x > 1.1\times10^{10}$, there ...
Mayank Pandey's user avatar
3 votes
1 answer
6k views

Why is this Goldbach's Conjecture Proof Wrong?

and I am sorry for the self-serving post, but I recently formulated a proof on Goldbach's Conjecture, and I was wondering why it was wrong. Let n be an even natural number greater than two, $n \in$ $...
user137451's user avatar
1 vote
1 answer
259 views

Can one prove a special case of Goldbach conjecture without constructing primes?

I know that sometimes in mathematics one can prove that there exist something without constructing it. I was thinking whether one can show if $2^{57885162}$ is a sum of two primes by any reasoning. ...
Student's user avatar
  • 185
2 votes
0 answers
305 views

On Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq N}{...
user135977's user avatar
5 votes
1 answer
368 views

Question about the proof of Goldbach's weak conjecture

H.A. Helfgott recently proved Goldbach's weak conjecture here: http://arxiv.org/pdf/1305.2897v2.pdf In (1.1), he explains that he is trying to show that $$\sum_{n_1 + n_2 + n_3 = N}\Lambda(n_1)\...
Mayank Pandey's user avatar
5 votes
1 answer
878 views

The relationship between Golbach's Conjecture and the Riemann Hypothesis

My question pertains to two famous groups of related conjectures: Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). ...
Doug's user avatar
  • 306
1 vote
1 answer
390 views

Is this statement equivalent to Goldbach's conjecture

Given a number $n\ge 3$, then one of these is true: \begin{equation} \begin{cases}2n = (6m-1)+P, \ \ \ P \in \mathbb P, \ 6m-1 \in \mathbb P, \ 6m+1 \in \mathbb P \ \ \ \ (1) \\ 2n-1 \in \mathbb P, \ ...
Valtteri's user avatar
  • 3,154
2 votes
1 answer
2k views

about Goldbach conjecture

About Goldbach conjecture, (that Every even integer greater than 2 can be expressed as the sum of two primes) and if algorithms exist to solve the Halting Problem, then algorithm that determine ...
user2939732's user avatar
7 votes
1 answer
2k views

Weaker Version of "Goldbach's Other Conjecture"

Taken from problem 46 on Project Euler: It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. $9 = 7 + 2 \times 1^2$ $15 = 7 + ...
Ryan's user avatar
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5 votes
1 answer
602 views

Binary vs. Ternary Goldbach Conjecture

Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current techniques?...
Andy Teich's user avatar
  • 1,555
1 vote
0 answers
3k views

Goldbach Conjecture Consequences

I have been looking into the Goldbach Conjecture pretty recently and I have often heard that it would have far-reaching consequences. However, I haven't found many of the actual consequences. I was ...
Jebruho's user avatar
  • 1,700
11 votes
3 answers
2k views

Disprove the Twin Prime Conjecture for Exotic Primes

The List of unsolved problems in mathematics contains varies conjectures of exotic primes like: Mersenne primes (of the form $2^p - 1$ where $p$ is a prime, A000668, $43\%$) Sophie Germain primes ($p$...
draks ...'s user avatar
  • 18.4k
4 votes
0 answers
738 views

A different approach to the strong Goldbach conjecture?

Consider the set A of prime numbers $p_i$ such that $p_i+6$ is not prime (listed in OEIS 140555; see comments thereto). Let 'Goldbach representation' mean a pair of odd prime numbers which sum to a ...
Keith Backman's user avatar

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