Given integers a,b,c,d∈[2n,2m] with m>n>1, how many primes p are there in [nα,nβ] for some 1<α<β such that 0<amod 0<b\bmod p<n^{\alpha/k} 0<c\bmod p<n^{\alpha/l} 0<d\bmod p<n^{\alpha/l} holds where k,l>2 is fixed? Assume n,m,\alpha,\beta,k,l are fixed. Heuristically we can solve for average case of choice of a,b,c,d following way: \frac{|\{p:a\bmod p<n^{\alpha/k}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{|\{p:b\bmod p<n^{\alpha/k}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{n^{\alpha/k}}{(n^{\beta}-n^\alpha)/\log(n^{\beta}-n^\alpha)} \frac{|\{p:c\bmod p<n^{\alpha/l}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{|\{p:d\bmod p<n^{\alpha/l}\}|}{|\{p:p\in[n^\alpha,n^\beta]\}|}\approx\frac{n^{\alpha/l}}{(n^{\beta}-n^\alpha)/\log(n^{\beta}-n^\alpha)} So \mathsf{Prob}(a\bmod … Read more