the pMultiMagic Feature  

Given a magic object in its regular sense, the figure remains summing to the same number when all numbers are rasised to the same power. The figure is classified as pMultimagic when this is the case for all powers uptil the number p 

^{m}INMagic Hypercube {H[_{j}i] ε [1..m^{n}] 
_{l=0}∑^{m} (H[[_{j}i,_{q}0]<_{q}l>])^{k} = _{l=0}∑^{m} (H[[_{j}0,_{q}m1]<_{j}m1l,_{q}l>])^{k} = (_{l=0}∑^{m^n} i^{k}) / m^{n1} ; k = 1..p} = Fh_{k}(m^{n}) / m^{n1} The exact sum terms merely depicts all 1agonals and all nagonals to sum over. (see hypercube article) 

With help of tridigital equations this author found 80 families of order 8 bimagic squares 224 order 9 bimagic square families where found with help of bidigital equations by this author and known bimagic squares by Collisson and Benson Jacobi identified by used methods each "family" of order m squares holds (modd(m))!!/2 different squares by main nagonal permutation currently only order 25 bimagic cubes where found (by John R Hendricks) (see: Authors Webpages for details, database section for full listing) 

pMultiMagic sum Fh_{p}(m^{n}) / m^{n1} 
Faulhaber's Formula (1631) Fh_{p}(n) = _{j=0}∑^{n} i^{p} = (1/(p+1))_{k=1}∑^{p+1}(1)^{δ(k,p)} (^{p+1}_{k}) B_{p+1k} n^{k} 

Fh_{1}(n) = (1/2) (n^{2} + n) 

notes: δ(k,p) kroenecker delta (1 iff k = p; 0 otherwise) and B_{p} (Bernouilli numbers) given by B_{0} = 1; B_{p+1} = (1/(p+2))_{k=0}∑^{p} (^{p+2}_{k}) B_{k} 
pMultiMagic Theorems  

A few general theorems regarding the pMultimagic feature are given below  
addition of constant to each number 
The addition of a constant to each number in a pMultiMagic hypercube the hypercube remains pMultimagic 

Proof: _{i=0}∑^{m1}(x_{i}+c)^{p} = _{i=0}∑^{m1}_{j=0}∑^{p} (^{p}_{j})x_{i}^{j}c^{pj} = _{j=0}∑^{p}(^{p}_{j}) (_{i=0}∑^{m1}x_{i}^{j})c^{pj} = _{j=0}∑^{p}(^{p}_{j}) [Fh_{j}(m^{n}) / m^{n1}] c^{pj} where the fact is used that we started with a pMultiMagic hypercube Q.E.D. 

coincidentially proven: Fh_{p}(m^{n}) = _{j=0}∑^{p}(^{p}_{j}) Fh_{j}(m^{n}) [(m^{n}+1)/2]^{pj} <=> _{j=0}∑^{p1}(^{p}_{j}) Fh_{j}(m^{n}) [(m^{n}+1)/2]^{pj} = 0 for odd p, since the magic sum for odd p = 0 for the centralized number range and (m^{n}+1)/2 is the appropriate shifting constant Q.E.D. 

pMultiMagic Complementairy invariance theorem 
The pMultiMagic sum is invariant for the complementairy transformation s[i] > (m^{n}+1)  s[i] 

Proof: using the addition of constant theorem move the range to [(m^{n}1)/2 .. (m^{n}1)/2] thus the complementairy number of s[i] = s[i] thus transformed the sum of powers of the complement yield: _{k=0}∑^{m1} (s[i]])^{p} = (1)^{p} _{k=0}∑^{m1} s[i]^{p} for odd p these sums are 0 while for even p (1)^{p} = 1 and the equation holds (The odd p sums are 0 since _{i=(m^21)/2}∑^{(m^21)/2}i^{p}= ((1)^{p}+1)_{i=0}∑^{(m^21)/2}i^{p}, which is 0 for odd p and even for even p, this sum divided by m^{n1} is the magic sum) Q.E.D. 

coincidentally proven: selfcomplementairy sequence sums to hf_{p}(m^{n})/m^{n1} sum when raised to an odd power p 

pMultiMagic Multiplication theorem 
The basic multiplication of pMultiMagic hypercubes is pMultimagic  
Proof: (A_{m} * B_{q})_{j}i = A_{j}i%m (B_{q,j}i/m  1)m^{n} _{i=0}∑^{mq1}(A_{j}i%m (B_{q,j}i/m  1)m^{n})^{p} = _{i=0}∑^{mq1}_{k=0}∑^{p}_{l=0}∑^{pk} (^{p}_{k})(^{pk}_{l}) (A_{j}i%m^{k} (B_{q,j}i/m^{l}(1)^{kl}m^{n}) notice that the k and l sums are summing over power < p since A and B are pMultiMagic these sums are constant on all the considered lines, hence the product is pMultiMagic Q.E.D. 