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}