# 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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### 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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### 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 ...
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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 ...
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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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### 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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### 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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### Is it possible that every $P$ as a prime number, can be expressed as a prime factor of $E$ such that $E$ is the sum of a pair of twin primes?

Curious about the Goldbach conjecture, and reading about twin primes, I was wondering if it is possible that every prime number as $P$, can be expressed as a prime factor of at least one $E$ such that ...
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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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### Goldbach Conjecture and the Busy Beaver function?

Background: Math undergrad, but complete layman in computer science I recently asked this question on CS stackexchange. I hope I am interpreting the answers correctly: Suppose we make a program to ...
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### 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 ...
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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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### Can any odd positive integer greater than $3$ be expressed as the sum of $2$ perfect square numbers (excluding $0$) plus $1$ prime number?

I have been reading about the "odd Goldbach conjecture" which states that: Every odd integer greater than $7$ can be written as the sum of three odd primes I have also been reading about the ...
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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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### 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?...
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### If an arithmetic function is multiplicative, non-zero at a prime, and "prime-linear", is it the identity?

Let $f:\mathbb{N}\to\mathbb{N}\cup\{0\}$ be a function. Let $f(1)=1,$ and $f(ab)=f(a)f(b)$ whenever $\gcd(a,b)=1.$ Note that I am assuming that $f$ is multiplicative but not completely multiplicative. ...
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### An "almost all" result for the binary Goldbach problem

I have a question. My professor in the lecture said that Vinogradov's method by applying the Hardy-Littlewood circle method (minor and major arc) for the ternary Goldbach problem can be used to prove ...
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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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### Goldbach's conjecture if 1 is counted as prime

I was grading some homework from a Survey of Mathematics course. They were asked to verify that Goldbach's conjecture holds for the first 15 even numbers greater than or equal to 4. A couple of ...
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I noticed that a slightly modified version of Goldbach's conjecture seems to hold for the quadratic $x^2+1$. Specifically, I assert for any even $n\geq 4$, there exists at least one pair \$p,q\in\...