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Questions tagged [twin-primes]

For questions on prime twins.

A twin prime is a prime number that differs from another prime number by two, for example the twin prime pair $(41, 43)$. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.

It is currently unknown whether or not there are infinitely many pairs of twin primes. However, it is known (Brun's theorem) that the sum of the inverses of the twin primes is finite (it is about $1.902$).

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Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is…
Tomas Wolf
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The significance and acceptance of Helfgott’s proof of the weak Goldbach Conjecture

Recently I was browsing math Wikipedia, and found that Harald Helfgott announced the complete proof of the weak Goldbach Conjecture in 2013, a proof which has been accepted widely by the math community, but according to Wikipedia hasn’t been…
D.R.
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Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who can explain the proof? Is the outline in the annals…
Trancot
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$n+1$ and $n \phi (n) + 1$ are both perfect squares if and only if $n$ is a product of twin primes?

I'm trying to prove the following conjecture concerning twin primes and Euler's totient function, which I have verified for $n$ up to 1 billion. For all $n \in \mathbb{N}$, $n+1$ and $n \phi (n) + 1$ are both perfect squares if and only if $n$ is a…
JMP
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Why is factorization of large number hard

Why factoring a number is difficult compared to finding out if it is prime (which can be done in polynomial time) ? I would think they might be of similar difficulty in terms of computational complexity.
arbitUser1401
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Whats wrong with this model theoretic proof of the twin primes conjecture?

I have a proof of the twin primes conjecture using the compactness theorem. It cannot be correct, because it is too simple. Please help find the flaw. Proof by contradiction, Assumption: there are only $n$ twin primes. Let $L$ be the language $\{ +,…
Max Kubierschky
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For primes $P_1$ and $P_2$, exists a prime $P_3$ that both $P_i + 6P_3$ is a prime

I was thinking about twin primes and I came to ask this question: If we have two distinct primes $P_1$ and $P_2$ which are both greater than $3$, then does there always exist a prime $P_3$ such that $$\begin{cases}P_1+6 P_3 \in \mathbb P \\ P_2+6…
Valtteri
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Is it possible to map infinitely many primes to infinitely many other primes with a linear function?

I was wondering if this is even posible because I can not see an answer... Given a subset $S\subset P$ from the set of prime numbers $P$. Let $S$ have an infinite number of elements. Is there any subset $S$ and a polynomial $Q(x) = ax + b$ on…
Ismael
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Twin Primes (continued research)

This has become increasingly crowded, so at the onset, let me state this: My question is, is there some reason this is so linear that I'm not seeing? The only thing it seems to indicate to me is that there truly must be infinitely many twin…
Elem-Teach-w-Bach-n-Math-Ed
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Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly improved and, according to the Polymath8 project home…
User X
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Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?

Thinking about Goldbach conjecture, I have the following question: Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes? For example, as 31 and 17 belong to the set of twin…
mathspika
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Modified Euler's Totient function for counting constellations in reduced residue systems

I am working on a modified totient function for counting constellations in reduced residue systems for the same range that Euler's totient function is defined over. This post is separated into three parts: "background" describes how this function…
Adam
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Twin prime conjecture proof error

I am absolutely sure this is wrong but I can't find why. For every integer $n$ there exist a finite number of primes less than $n$. Take the set containing those primes and multiply them together to get $x$. Aren't $x+1$ and $x-1$ prime, implying…
shai horowitz
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What is notable about the composite numbers between twin primes?

Look at the composites between twin primes (A014574): $$ 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, \\ 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, \ldots \;. $$ Is there anything special about their…
Joseph O'Rourke
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Sets of Prime and Composite Numbers

We know that all primes are of the form $ 6k ± 1 $ with the exception of 2 and 3. We also know that not all numbers of the form $ 6k ± 1 $ are prime. This leads to four distinct sets of pairs adjacent to a multiple of six: Twin Primes, Example: $…
Tony
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