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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.

The Goldbach conjecture originates from a $1742$ letter from Goldbach to Euler, states that every even integer greater than $3$ is the sum of two primes. The conjecture has been shown to hold for all integers less than $4 × 10^{18}$, but remains unproven despite considerable effort.

Goldbach's weak conjecture is that every odd integer greater than $6$ is the sum of three 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 undecidable, there is no proof it is false. If we…
Sriotchilism O'Zaic
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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 conjecture is an undecidable proposition.
Yes
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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 prime if its only positive integer divisors are…
klorzan
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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 myself to understand those papers ? Maybe there will…
jim
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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 function, error terms of Prime Number Theorem, and Farey…
david
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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 $11$…
Damian Reding
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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. I tried to verify this up to some extent.
albo
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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 $D=N-P$. Then $D$ is always a prime. (Except $D=1$)" Can…
al-Hwarizmi
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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 ($p$ and also $2p+1$ is prime, A005384,…
draks ...
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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 $p$ can be expressed as the sum of $p$ primes where…
Jamal Farokhi
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Prove or disprove: $\forall a \ne k^2, \exists b, a^3 - b^2 \in \mathbb{P}$

In a forum post the following conjecture was proposed with no background information. For all positive non-perfect-square number $a$, there exists integer $b$ such that $$ a^3 - b^2 \text{ is a prime number.} $$ By prime numbers, here we only take…
PinkRabbit
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Conjecture: All but 21 non-square integers are the sum of a square and a prime

Update on 6/19/2020. This discussion led to deeper and deeper results on the topic. The last findings are described in my new post (including my two answers), here. I came up with the following conjecture. All non-square integers $z$ can be…
Vincent Granville
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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 it is true and assuming that it is false would…
abcd8888
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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 occurrence of a certain kind of prime gap, but I don't…
Mario Carneiro
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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 Vaughan showed that "most" even numbers were…
martin
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