Simple: The only prime is $5.$ We consider $\mod 2.$ Clearly $p>2.$ $ x+y=p, z-w=p$ where $x,y,z,w,p$ are primes. Now, by $\mod 2$ we get that only one from $\{x,y\}$ is $2.$ Say $x.$ Similarly, by $\mod 2$ we get that only one from $\{x,y\}$ is $2.$ Clearly it's $w.$ So we get $x, p, z$ is AP with $d=2.$ By $\mod 3$ we get $(x,p,z)=(3,5,7).$ So the only prime is $5.$

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