Maybe an idiot question but I can't find any info! We divide successive prime numbers by some fixed prime number $n$ (e.g. 7 or 17). We'll get some remainders $r[i]= 1..n-1$ Is there any law or theorem about their distribution? It seems Fermat's Little theorem and Chinese remainder theorem don't work.. I've tried it in Mathematica and it seems remainders are chaotic. But "Poincaré 3D view" $\{ r[i],r[i-1],r[i-2]\}$ shows some lines and nets.
UPD Thanks to everybody, esp. TonyK! It seems the answer is:
Let $\mathbb{P}(d)$ - probabilty for distance between successive primes to be $d$. (It depends on value of "first" number and known only numerically). If some prime number $p_1$ has reminder $r_1$ when divided by $n$, than probability for the next prime $p_2$ to have reminder $r_2$ is: $\sum_{k=0}^{\infty} \mathbb{P}(r_2-r_1+k \cdot n)$