For container structures (such as tables, rtables) use e.g. EqualEntries  (or LinearAlgebra:-Equal(...), or ...)

EqualEntries(Matrix([[1,2],[1,2]]), <1,2;1,2>);

        true

The equation has a unique real solution, so, "Diophantine" is not essential.

restart;
f:=419*x^2 + 116*x*y - 426*x*z + 78*y^2 - 142*y*z + 133*z^2 - 1604*x - 682*y + 1086*z + 2306:
CS:=Student:-Precalculus:-CompleteSquare:
CS(f,z):
f1:=select(has,[op](%), z)[]:
CS(f-f1,y):
f2:=select(has,[op](%), y)[]:
f3:=CS(expand(f-f1-f2),x):
expand(f-f1-f2-f3);
solve([f1,f2,f3]);

          0
          {x = 7, y = 11, z = 13}

By Dirichlet's theorem on arithmetic progressions - Wikipedia there are infinitely many primes of the form 24*n + 1.
p,q can be any pair of these.

P.S. We could avoid Dirichlet' theorem noting that at least one of the sets {24*n+k| n in N} for k=0,1,...,23   must contain infinitely many primes.

You want the limit of V for beta --> oo. Use:

limit(V, beta = infinity)

It results 6.5.

If you want to check this numerically for large values for beta (as you did), increase first Digits e.g.

Digits := 30;

P.S. Note that usually, for symbolic computations like limit, it is recommended to use exact values or rational values for parameters.

The extra question is not clear enough (for me).

Download periodic-vv.mw

Maple does not use LeafCount for the complexity.

length~([e1,e2]);  # same complexity

                     [21, 21]

Better use ifactors:

SumPrimeFactors := n -> add(map2(op,1,ifactors(n)[2])):

Your matrix m  is positive semidefinite. But if we compute M := evalf(m) it will become indefinite (due to roundoff errors.)

On the other hand for such M  and a given permutation matrix P, the LDLt factorization of P^+ .M.P may not exist, so the behavior you describe is normal.

The error given by Maximize is clear:
Error, (in Optimization:-NLPSolve) no feasible point found for the linear constraints

Indeed, C_1, and C_2 imply:

C12:=op(C_1 union C_2):
solve(C12[1]);solve(C12[2]);

                  [36571.46530, infinity)

               (-infinity, 27143.30624]

so, disjoint intervals, no common point (i.e. no feasible point).

Re(term2) assuming real;
combine(%);

Download inf-prod-vv.mw

A table has special evaluation (to a name). So, use:

eval(T1["1"]);

Also, print(T1["1"]); confirms.

See the 2018 post  add, floats, and Kahan sum - MaplePrimes.

Note that since 2021, the add command has the pairwise option, see Pairwise summation - Wikipedia.