Intereting Posts

Properties of the element $2 \otimes_{R} x – x \otimes_{R} 2$
Evaluate $\int_0^1\frac{x^3 – x^2}{\ln x}\,\mathrm dx$?
The limit of the alternating series $x – x^2 + x^4 – x^8 + {x^{16}}-\dotsb$ as $x \to 1$
Metrizability of a compact Hausdorff space whose diagonal is a zero set
How were the sine, cosine and tangent tables originally calculated?
showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
Important applications of the Uniform Boundedness Principle
Contradicting Fubini's theorem
weak homotopy equivalence (Whitehead theorem) and the *pseudocircle*
How can I prove that the additive group of rationals is not isomorphic to a direct product of two nontrivial groups?
How do I find the Intersection of two 3D triangles?
prove equality with integral and series
In Russian roulette, is it best to go first?
Exponentiation as repeated Cartesian products or repeated multiplication?
Definition of Conditional expectation of Y given X.

In the context of sieving for twin primes ($p\#$ is the primorial function) the following seems true.

The number of $n$ such that $n, (n+2)$ are coprime to $p_k\#$ for $n=1,2,…,~p_k\#$ is $\prod_{i=2}^k (p_i-2).$

Calculation of the first 7 products and a bit of luck in OEIS suggest this is true but it does not seem so easy to prove. I notice that $n+2$ is a simple permutation of $GCD(n,p_k\#)$ and so maybe using characters? Partly I am posting this because I haven’t seen it before so someone else might find it interesting. It does not lead (directly) to a useful sieve because $p_k\#$ grows too fast but probably it has been considered somewhere.

- Probability of rolling three dice without getting a 6
- Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K
- Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$
- How to algebraically prove $\binom{n+m}{2} = nm + \binom{n}{2} + \binom{m}{2}$?
- Proving binomial coefficients identity: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$
- Counting all possible legal board states in Quoridor

- product of six consecutive integers being a perfect square
- How many combinations can I make?
- What's a BETTER way to see the Gauss's composition law for binary quadratic forms?
- An identity involving the Pochhammer symbol
- Proof of Wolstenholme's theorem
- Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
- counting monotonically increasing functions
- Are $121$ and $400$ the only perfect squares of the form $\sum\limits_{k=0}^{n}p^k$?
- A certain unique rotation matrix
- How many planar arrangements of $n$ circles?

Let me consider $p_k$ primes.

According to the Chinese Remainder Theorem

every number $n$ from $1$ to $p_1p_2\cdots p_k$ is uniquely determined by

the set of its $k$ remainders $r_i$ when $n$ is divided by $p_i$.

In practice to every choice of remainders $r_i$ corresponds one and only one number $n$ from $1$ to $p_1p_2\cdots p_k$.

So, we can ask, in how many ways can we choose remainder $r_i$ ? We want it different from 0, otherwise $n$ is divisible by $p_i$ and we want it different from $p_i-2$, otherwise $n+2$ is divisible by $p_i$.

So, apart the case of $p=2$ in which only the remainder 1 is admissible,

in all the other cases there are $p_k-2$ possible remainders.

Hence, the number of sets of “good” remainders (and thus the number of “good” pairs) is just

$$(p_2-2)(p_3-2)\cdots (p_k-2)

$$

as you imagined.

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