DICKSON CONJECTURE PROOF

In 1904, Dickson [6] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [3]generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to themultivariable case. In 2006, Green and Tao [9] considered Dickson’sconjecture in the multivariable case and gave directly a generalizedHardy-Littlewood estimation. But, the precise Dickson’s conjecture inthe multivariable case does not seem to have been formulated. In thispaper, based on the idea in [8] a partial proof of Dickson's Conjecture is provided .Let $\{a_ {1},a_{2},....a_{k}\}$ the set of $k$ linear prime admissible , $t \geq 1$, $q_{a_{t}}$ be the smallest prime number dividing $a_{t}$ and $\omega(q_{a_{t}})$ its order by arranging the prime numbers in ascending order.$\beta_{j}(\sqrt{n})$ the number of prime $p\leq \sqrt{n}$ such that $ a_{j}p+b_{j}$ is prime .Let \begin{eqnarray}G(\omega(q_{a_{t}}))=\left[ \frac{1}{\phi(a_{t})}+ \frac{ 1}{q_{a_{t}}\phi(a_{t})} -\frac{1+q_{a_{t}}}{q_{a_{t}}\phi(a_{t})}\prod_{i=1}^{\omega(q_{a_{t}})-1}\left[ 1-\frac{1}{p_{i}^{\sigma^{-1}(i)}(p_{i}-1)} \right]\right]\\R(r,t)=\frac{1}{\phi(a_{t})}\left[1-\prod_{i=\omega(q_{a_{t}})+1,p_{i}\mid a_{t}}^{r}\left[ 1-\frac{1}{p_{i}^{\sigma^{-1}(i)}} \right]\prod_{i=\omega(a_{t})+1,p_{i}\nmid a_{t}}^{r}\left[ 1-\frac{1}{p_{i}^{\sigma^{-1}(i)-1}(p_{i}-1)}\right]\right]\\\mu(k,r) = \sum_{t=1}^{k}\Pi(a_{t}n+b_{t})\prod_{i=1}^{r}\left[ \frac{\prod_{p\mid a_{i}}p^{v_{p}(a_{i})}p_{i}-1}{\prod_{p\mid a_{i}}p^{v_{p}(a_{i})}p_{i}}\right]\end{eqnarray}Let $ H(n)$ the number of prime $p$ less that $n$ such that :$ \forall i \leq k,a_{i}p+b_{i}$ is prime and $ Q(n)$ the number of prime such $\exists i \leq k ,a_{i}p+b_{i}$ is primeWe show that :\begin{eqnarray}H(n)-Q(\sqrt{n})\sim_{+\infty }\Pi(k,n)-\mu(k,r)\\Q(n)-Q(\sqrt{n})\sim_{+\infty }\Pi(k,n)-\sum_{t=1}^{k}\Pi(a_{t}n+b_{t})\left[G(\omega(q_{a_{t}}))+ R(r,t)\right]\end{eqnarray} Where $ \Pi(k,n)=\Pi(\min(a_{1},a_{2},..a_{k})n+\max(b_{1},b_{2},..b_{k}))$ \end{center}

INFINITUDE OF PRIME p such that ap+b is prime where a, b are coprime integers

In 1904, Dickson [6] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [3]generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to themultivariable case. In 2006, Green and Tao [9] considered Dickson’sconjecture in the multivariable case and gave directly a generalizedHardy-Littlewood estimation. But, the precise Dickson’s conjecture inthe multivariable case does not seem to have been formulated. In thispaper, based on the idea in [8] we introduce an interesting class of prime numbers to solve the dickson conjecture Although this article does not solve the dickson conjecture but it solves a problem that is similar to the Dickson conjecture. the problem is stated as follows being given two coprime integers a, b there is an infinity of prime numbers p such that ap+b is prime. This type of prime numbers we call it Bado-Tiemoko prime numbers .We intend to generalize this result but for the moment we speculate that given a family $(a_{i},b_{i})_{1\leq i\leq k}$ such that $\gcd(a_{i},b_{i})=1,\forall 1\leq i\leq k$ there is an infinity of prime numbers $p$ such that $a_{i}p+b_{i}$ is prime for $\forall 1\leq i\leq k$ Let $q_{a}$ be the smallest prime number dividing $a$ and $\omega(q_{a})$ its order by arranging the prime numbers in ascending order.Let $\beta(n)$ the number of Bado-Tiemoko prime less than $n$ and $$\eta_{s,i}=\frac{(p_{i}-1)^{\delta_{p_{i}}(a)}\prod_{k=1}^{s}(p_{i_{k}}-1)^{\delta_{p_{i_{k}}(a)}}}{\phi(a)\prod_{k=1}^{s}(p_{i_{k}}-1)(p_{i}-1)\prod_{k=1}^{s}p_{i_{k}}^{\delta_{p_{i_{k}}(a)}}p_{i}^{\delta_{p_{i}}(a)}}$$ $$\mu(r)=\frac{1}{\phi(a)}[\sum_{s=1}^{\omega(q_{a})}(-1)^{s-1}\sum_{1\leq i_{1}

DADU SICHERMAN (SUATU APLIKASI DARI FAKTORISASI TUNGGAL PADA Z[X])

An interesting application of unique factorization in Z[X] is Sicherman dice. The dice is a pair of dice whose has different number from ordinary dice which faces are labeled 1 through 6. But probability the sum of faces are same as the sum of ordinary dices. Sicherman dice is obtained by using one to one correspondence between the two polynomials and the face of two dice of ordinary dice.Kata Kunci : ring R[X], daerah Integral, polynomial irreducible, Faktorisasi tunggal pada Z[X].

An application of elementary real analysis to a metabelian group admitting integral polynomial exponents

Let