Given two distinct primes $P\_1$ and $P\_2$ picked randomly and uniformly in the interval $[T^2,2T^2]$ consider the set $\chi(P\_1,P\_2)$ of numbers of form $$xP\_1-yP\_2$$ where $x,y$ are in $[0,T^{1+\epsilon}]$ where $T>0$ and $\epsilon\geq0$ is small.



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> Are there expected to be at least $\frac{T^\mu}{O(\log T)}$ many distinct primes in the set $\chi(P\_1,P\_2)$ which are of size $T^{1+\mu}$ for small $\mu\geq0$?
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> If not what is the correct asymptotic at least under appropriate assumptions?
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Assume $T^{\min(\mu, \epsilon)}=\omega(\mathsf{polylog}(T))$.



Expected number of primes of particular size and from a linear form
The set $\chi(P\_1,P\_2)$ has $T^{2+2\epsilon}$ distinct elements in $[-2T^{3+\epsilon},T^{3+\epsilon}]$ (i.e., density of order $T^{\epsilon-1}$),
so we would expect $\chi(P\_1,P\_2)$ to contain about $T^{\mu+\epsilon}$ elements in $[C\_1T^{1+\mu}, C\_2T^{1+\mu}]$ provided that $0<C\_1<C\_2$. Of these, a fraction of about $1/(\log T)$ should be primes, leading to a prediction of about
$$\frac{T^{\mu+\epsilon}}{\log T}$$ primes in $[C\_1T^{1+\mu}, C\_2T^{1+\mu}]$ .


