# Questions tagged [goldbachs-conjecture]

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

141 questions
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### Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture

In the most recent numberphile video, Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows: If the hypothesis is ...
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### 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 ...
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### Proof: 1007 can not be written as the sum of two primes.

The claim is: 1007 can be written as the sum of two primes. We want to prove or disprove it. Edit: My professor provided this definition in his previous assignment: An integer $n \geq 2$ is called ...
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### 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 ...
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### $2n = \phi(a) + \phi(b)$

The values of the Euler phi function $\phi(n)$ are tabulated at OEIS A$000010$. Each of these values is even except for $\phi(1) = \phi(2) = 1$ . However, not every even number arises in this way. ...
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### Useful reformulation of Goldbach's conjecture?

Let us assume there exists some infinite order differential equation whose solution is: $$y= \sum_{n=1}^\infty A_n \exp(p_n^sx)$$ Where $p_n$ is the $n$'th prime. Substituting $y=\exp(\lambda x)$...
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### Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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### How could it be possible for the Goldbach conjecture to be undecidable?

The answer to the question "Could it be that Goldbach conjecture is undecidable?" claims that it is possible for something such as the Goldbach conjecture to be undecidable, meaning that assuming that ...
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### 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 ...
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### 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 ...
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### 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. $9 = 7 + 2 \times 1^2$ ...
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### Weakening Goldbach hypothesis by allowing finitely many composites and 1

Is it an open question whether there is a finite set $N$ of positive integers such that for every positive even integer $n$ there are $n_1,n_2\in\mathbb P\cup N$ such that $n=n_1+n_2$? ($\mathbb P$ ...
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### For any $a \in \Bbb{Z}$, can we always find two prime numbers $p, q$, such that $p - q \in (a)$? [duplicate]

This is a major weakening of many prime sum / difference existence questions. Let $a \in \Bbb{Z}$ and $(a)$ the ideal generated by $a$. Then do there exist two primes $p, q$ such that $p - q \in (a)$...
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### Is every sufficiently large even integer the sum of distinct primes?

Is every sufficiently large even integer the sum of (any number of) distinct primes? No doubt this question has been asked before; does the conjecture/theorem have a name? It is related to Goldbach'...
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### Taking an unproven but “seemingly true” statement as an axiom

I'm sorry for this uneducated question, but I've been thinking of this for a few hours and I couldn't find anything on the topic. Perhaps it is just a failure on my part and a limitation of my ...
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### Is a strong form of Goldbach conjecture equivalent of Generalized Riemann Hypothesis?

In Andrew Granville's paper: REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS He said that: "we show that if a strong form of Goldbach's conjecture is true then every ...
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### Sum and Product Puzzle and Prime Factors

Suppose we have two number $X$ and $Y,$ such that $1 < X < Y < 100,$ and $X + Y ≤ 100.$ Sue is given $S = X + Y$ and Pete is given $P = XY.$ They then have the following conversation: Pete:...
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### 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 ...
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### 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). ...
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### Two new conjectures related to Lemoine's and Goldbach's

Lemoine's conjecture can be written: $2n + 1 = p + 2q$ always has a solution in primes $p$ and $q$ (not necessarily distinct) for $n > 2$. It appears that the following very similar conjectures (...