I am really interested in sufficient conditions on $a\_i, b\_i$ guaranteeing that the linear forms $a\_1\phi(n)+b\_1,\dots, a\_k\phi(n)+b\_k$ become simultaneously prime for infinitely many positive integers $n$. Here, $\phi(n)$ is the Euler's totient function and $a\_i, b\_i$ are arbitrary positive integers. 


**Background.** In the 1904 paper of L. E. Dickson titled *A new extension of Dirichlet’s theorem on prime numbers, Messenger of Math. 33 (1904), 155–161*, 
the problem of if the forms $a\_1 n+b\_1 ,\cdots ,a\_kn+b\_k$ become simultaneously prime infinitely many times is discussed (See [Dickson's Conjecture](https://en.wikipedia.org/wiki/Dickson%27s_conjecture).) In my question here $n$ is replaced with $\phi(n)$.


**Note**: The motivation of this question is the estimation of primes using Euler's toitient function and also to know if there is any connection between Lucas-Lehmer Primality test and the titled affine-linear form for $d=1$ and $t=1$ because the latter depend on relationship between $n$ and $\phi(n)$ 



Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?
Most probably $\gcd(a\_i,b\_i)=1$ and $a\_i \geq 1$ and $b\_i$ to be odd for $i=1,...,k$ is enough.


Since $\{\varphi(n): n \in \mathbb N\}$ can be partitioned into countably many subsets $C\_l=\{\varphi(w\_{rl}): r \in \mathbb N\}$ such that $\varphi(w\_{ml})<\varphi(w\_{nl})$ if $m<n$ the $k$ forms $a\_j \cdot \varphi(n) +b\_j$ for $j=1,...,k$ are all prime for an infinite number of values of $n$ if they are:


a) prime for at least one $r$ for every $l$


b) prime for an infinite number of $r$˙s for some $l$


The b) leads to research of subsequences of $\mathbb N$ such that Dickson´s conjecture is true for those subsequences, and, I think that even in the case $k=1$ that is not fully resolved.


In other words, the necessary and sufficient conditions for a subsequence $s\_{\alpha(i)}$ of $\mathbb N$ to give an infinite number of primes $as\_{\alpha(i)}+b$ for $\gcd(a,b)=1$ can be stated, but the generalization of Dirichlet´s theorem given in this form, is, I think, not yet available anywhere, in other words, not proven anywhere.


And you ask even more, that is, slight modification of generalization of Dickson´s conjecture, and, there is not much that can be surely told about that, since even the Dickson´s conjecture is unsolved.


