A Difficult Application of Modular Arithmetic

Question

Looking to input a $p$ for the first line:

p=6;

Build a list of primes from the 3rd to $p-1$:

plist=Table[Prime[n],{n,3,p-1}];

Then build a list as follows:

bmod=Outer[Mod,{3*Prime[p]},plist]

---

Here's the code so far with the output:

p=6;
plist=Table[Prime[n],{n,3,p-1}];
bmod=Outer[Mod,{3*Prime[p]},plist]

{{4,4,6}}

---

Now I'd like to multiply each element of the list by $2$, but keep them mod their original prime, so instead of:

{{8,8,12}}

I get:

{{3,1,1}}

Then, I'd like to add this list to the original, still keeping the modular bounds:

{{3,1,1}}
+
{{4,4,6}}
=
{{2,5,7}}

And then repeat this process several times, always adding to the previous line:

{{4,4,6}}
{{2,5,7}}
{{0,6,8}}
{{3,0,9}}
etc.

Assuming you've followed my so far, how would I do that?

Answer 1

Starting with

p = 6;
plist = Table[Prime[n], {n, 3, p - 1}]
bmod = Outer[Mod, {3*Prime[p]}, plist] // Flatten
(* {5, 7, 11} *)
(* {4, 4, 6} *)

the first result is obtained by

Thread[Mod[2 bmod, plist]]
(* {3, 1, 1} *)

To obtain the next results, use

NestList[Thread[Mod[# + bmod, plist]] &, %, 5]
(* {{3, 1, 1}, {2, 5, 7}, {1, 2, 2}, {0, 6, 8}, {4, 3, 3}, {3, 0, 9}} *)

which adds bmod to the previous result, modulo the corresponding prime in plist.

Addendum

To test whether any of the results immediately above contain 2 or 4, as requested in a comment, use

MemberQ[#, (2 | 4)] & /@ %
(* {False, True, True, False, True, False} *)