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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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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&...
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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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Gap between an even integer and the next smaller prime?

I am desparately searching for a case that would skip the following conjecture (a variation of the Goldbach conjecture): "Let $N$ an even integer, $P$ the very next prime smaller than $N$, and $D=N-P$...
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Can every even integer greater than four be written as a sum of two twin primes?

Thinking of Goldbach conjecture I arrived at this $\mathrm{Conjecture}$: Every even integer greater than four can be written as a sum of two twin primes. What do you think? I hope this is true. ...
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Is this extension of Goldbach's conjecture obviously false?

Goldbach's conjecture is: Every even integer greater than $2$ can be expressed as the sum of two primes. Extension of Goldbach's conjecture is: Every number from $p\mathbb{Z}$ greater than $p$ ...
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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 ($p$...
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Goldbach for certain classes of $n$

The Wiki article on the Goldbach conjecture (where $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$) states that In 1975, Hugh Montgomery and Robert Charles ...
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Goldbach's conjecture with negative primes

Is the Goldbach conjecture any easier if we allow primes to be negative as well? That is, every even integer is the sum or difference of two primes. The twin prime conjecture talks about the ...
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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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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 ...
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Is every large enough even integer the sum of two quasiprimes ?

Call a "quasiprime" any integer of the form $ p^{k} $ with $ k\geq 1 $ and $ p $ a prime number. In other words, the set of quasiprimes is exactly the set of integers for which the von Mangoldt $ ...
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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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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)\...
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A heuristic argument for the Goldbach conjecture?

This question here is purely speculative so be warned if you read on: This question is related to a sequence $b_n$ which is defined here: A series related to prime numbers For the numbers $a_{2n,2}$ ...
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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 ...
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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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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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How to turn number into sum of unique primes?

I have to find algorithm which find prime number less than $n$ which is sum of the largest amount of unique primes, for example for $n=81$, the answer is $79 = 3 + 5 + 7 + 11 + 13 + 17 + 23$. I have ...
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Special case of Goldbach-conjecture

Is every number $2p$ , $p>3$ prime , the sum of two DISTINCT primes ? The case $n=2p$ is no counter-example of the Goldbach-conjecture because $n=p+p$ shows that $n$ is the sum of two primes. ...
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solutions of $p = 2q + r$

let $P$ denote the rational primes of the for $4k+3$, and let $Q$ denote the set containing $1$ and all the rational primes of form $4k+1$. let $p \in P$. we look for representations of $p$ of the ...
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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}...
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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 ...
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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, $n \in$ $...
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On the number of representations of a positive integer into the form $x=p+dq$, where $p$ and $q$ are primes and $d$ is a given positive integer

I want to ask the estimate of $$P_d(x)=\text{card}\{ p\in\mathbb{P} \mid x=p+dq \ \ \text{ for some } q\in \mathbb{P}\},$$ where $d$ is a given positive integer(or a sufficiently small positive ...
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Are there any intuitive reasons for Goldbach conjecture to be true?

One thing puzzled me is that, despite its simple form, I have not seen any intuitive reasons for Goldbach conjecture to be true. Typical heuristic reason is based on probability arguments. Such ...
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Is every positive integer greater than $2$ the sum of a prime and two squares?

I'm not sure if this conjecture is less hard than Goldbachs conjecture: any integer greater than $2$ is the sum of an odd prime and two squares of integers. Facts as: Every prime of the form $4n+1$ ...
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Is there any algorithm to identify the smallest example of gap size $n-1$ between consecutive prime numbers

In contemplating Goldbach's conjecture, I became interested in gaps between successive primes. If $n<a<b<2n$ and the range $a$ to $b$ is a primeless gap, then one could ignore any primes in ...
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Why can't we prove Goldbach's conjecture with this method?

1: Prime gap bounds: Consider the following non-asymptotic bounds for $\pi(x)$, proven by Dusart in 2018 (holding for $x>5393$): $$\frac{x}{\log(x)-1}<\pi(x)<\frac{x}{\log(x)-1.112}$$ To ...
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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 (...
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Implications of disproving the Goldbach's Conjecture

What would be the most important implications of finding an even number that cannot be expressed as the sum of two primes? Would the existence on one such number in anyway predict the likeliness of ...
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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, ...