[proofplan] We prove the theorem by invoking the Green-Tao transference mechanism with a prime-supported $W$-tricked weight, so that positivity already means primality and not merely being a prime power. After passing to a residue class modulo a primorial $W$, the [prime number theorem](/theorems/1692) in arithmetic progressions gives positive density for this weight. The Green-Tao enveloping sieve supplies a finite majorant whose linear forms and correlation estimates, at the complexity required for $k$-term progressions, verify the hypotheses of the [relative Szemerédi theorem](/theorems/4610). Relative Szemerédi then gives a nonzero weighted count of $k$-term arithmetic progressions in the $W$-tricked primes, and undoing the $W$-trick converts one of them into an arithmetic progression of primes. [/proofplan]

[step:Fix the progression length and pass to a residue class modulo a primorial] Fix an integer $k \ge 1$. For $k = 1$, any prime gives the required progression, so assume $k \ge 2$. Let $w \in \mathbb{N}$ be a parameter that will tend to infinity, and define the primorial \begin{align*} W := \prod_{p \le w} p, \end{align*} where the product is over primes $p$. Define the prime-supported von Mangoldt weight \begin{align*} \Lambda_{\mathrm{pr}}: \mathbb{N} &\to [0,\infty) \\ m &\mapsto \begin{cases} \log m, & \text{if } m \text{ is prime}, \\ 0, & \text{otherwise}. \end{cases} \end{align*} For each residue class $b \in \{1,\dots,W\}$ with $\gcd(b,W)=1$, define the prime-supported $W$-tricked von Mangoldt function \begin{align*} f_{N,b,W}: \{1,\dots,N\} &\to [0,\infty) \\ n &\mapsto \frac{\varphi(W)}{W}\,\Lambda_{\mathrm{pr}}(Wn+b), \end{align*} where $N \in \mathbb{N}$ and $\varphi$ is Euler's totient function. The [prime number theorem](/theorems/1742) in arithmetic progressions implies that, for each fixed $W$ and each $b$ with $\gcd(b,W)=1$, \begin{align*} \frac{1}{N}\sum_{n=1}^{N} f_{N,b,W}(n) \to 1 \end{align*} as $N \to \infty$. [/step]

[step:Majorize the $W$-tricked primes by a pseudorandom enveloping sieve] We use the Green-Tao enveloping sieve theorem in the finite cyclic interval model needed for $k$-term arithmetic progressions. For the fixed integer $k$, choose the sieve parameters in the Green-Tao construction with linear-forms complexity and correlation order large enough for the relative Szemerédi theorem for $k$-term progressions. Concretely, for every fixed finite system $\Psi=(\psi_i)_{i=1}^{t}$ of pairwise non-proportional affine-linear forms \begin{align*} \psi_i: \mathbb{Z}^{d} &\to \mathbb{Z} \end{align*} whose number of forms $t$, dimension $d$, and integer coefficients are bounded by constants depending only on $k$, the majorant must satisfy the linear forms estimate \begin{align*} \frac{1}{M^{d}}\sum_{x \in \{1,\dots,M\}^{d}} \prod_{i=1}^{t} \nu_{N,b,W}(\psi_i(x)) = 1 + o_{N \to \infty}(1), \end{align*} whenever all values $\psi_i(x)$ lie in $\{1,\dots,N\}$. It must also satisfy the corresponding correlation estimates up to the bounded order required by the Cauchy-Schwarz expansion of the $k$-term progression count: products of at most that many translates of $\nu_{N,b,W}$ are bounded by a fixed correlation weight with bounded moments. These two estimates are the Green-Tao linear forms condition and Green-Tao correlation condition at level $k$. The enveloping sieve theorem supplies, after $w$ is chosen sufficiently large in terms of $k$ and then $N$ is chosen sufficiently large in terms of $w$ and $k$, a function \begin{align*} \nu_{N,b,W}: \{1,\dots,N\} &\to [0,\infty) \end{align*} with mean $1+o_{N \to \infty}(1)$ satisfying these two estimates uniformly in every residue class $b$ with $\gcd(b,W)=1$. It also gives the pointwise domination \begin{align*} 0 \le f_{N,b,W}(n) \le C_k\nu_{N,b,W}(n) \end{align*} for every $n \in \{1,\dots,N\}$, where $C_k>0$ is finite and depends only on $k$, not on $N$, $b$, or the admissible choice of $W$ after the above parameter selection. Therefore the normalized function built from $f_{N,b,W}$ is eligible for the Green-Tao relative Szemerédi theorem. [/step]

[step:Apply relative Szemerédi to obtain a progression in the $W$-tricked primes] Choose $N$ large enough that \begin{align*} \frac{1}{N}\sum_{n=1}^{N} f_{N,b,W}(n) \ge \frac{1}{2}. \end{align*} Define \begin{align*} g_{N,b,W}: \{1,\dots,N\} &\to [0,\infty) \\ n &\mapsto C_k^{-1} f_{N,b,W}(n). \end{align*} The domination estimate from the enveloping sieve gives $0 \le g_{N,b,W} \le \nu_{N,b,W}$, and the preceding density estimate gives \begin{align*} \frac{1}{N}\sum_{n=1}^{N} g_{N,b,W}(n) \ge \frac{1}{2C_k}. \end{align*} The Green-Tao relative Szemerédi theorem for $k$-term arithmetic progressions applies because its hypotheses are precisely: a non-negative majorant of mean $1+o(1)$ satisfying the level-$k$ Green-Tao linear forms and correlation conditions, a function bounded above by that majorant, and average at least a fixed positive density. These have just been verified for $\nu_{N,b,W}$ and $g_{N,b,W}$ with density parameter $\delta=1/(2C_k)$. Hence there exist integers $a,r$ with $1 \le a$, $1 \le r$, and $a+(k-1)r \le N$ such that \begin{align*} \prod_{j=0}^{k-1} g_{N,b,W}(a+jr) > 0. \end{align*} Since $g_{N,b,W}$ is a positive scalar multiple of $f_{N,b,W}$, this implies \begin{align*} \Lambda_{\mathrm{pr}}(W(a+jr)+b) > 0 \end{align*} for every $j \in \{0,\dots,k-1\}$. [/step]

[step:Undo the $W$-trick to recover a prime arithmetic progression] For each $j \in \{0,\dots,k-1\}$, the positivity of $\Lambda_{\mathrm{pr}}(W(a+jr)+b)$ implies directly from the definition of $\Lambda_{\mathrm{pr}}$ that $W(a+jr)+b$ is prime. Define \begin{align*} P_j := W(a+jr)+b \end{align*} for each $j \in \{0,\dots,k-1\}$. The sequence $(P_j)_{j=0}^{k-1}$ is an arithmetic progression because \begin{align*} P_j = (Wa+b) + j(Wr) \end{align*} for every $j \in \{0,\dots,k-1\}$. Its common difference is $Wr>0$, and it has length $k$. Therefore the primes contain an arithmetic progression of length $k$. [/step]