What is the n-dimensional (hyper)cube's total number of entities modulus 729?

Geometry Level 2

A n n -dimensional (hyper)cube has vertices (entity n , 0 {}_{n,0} ), edges (entity n , 1 {}_{n,1} ), faces (entity n , 2 {}_{n,2} ), etc. So that words do not have to be invented for each n n -dimensional entity type. I will be referring to them as entity n , m {}_{n,m} or entities n m {}_{n\,m} as appropriate Please, note that the full dimension entity is also counted. Here are some sample entity counts:

dimension n = 0 n = 1 n = 2 n = 3 n = 4 entity n , 0 1 2 4 8 16 entity n , 1 0 1 4 12 32 entity n , 2 0 0 1 6 24 entity n , 3 0 0 0 1 8 entity n , 4 0 0 0 0 1 \begin{array}{l|rrrrr} \text{dimension} & n=0 & n=1 & n=2 & n=3 & n=4 \\ \text{entity}_{n,0} & 1 & 2 & 4 & 8 & 16 \\ \text{entity}_{n,1} & 0 & 1 & 4 & 12 & 32 \\ \text{entity}_{n,2} & 0 & 0 & 1 & 6 & 24 \\ \text{entity}_{n,3} & 0 & 0 & 0 & 1 & 8 \\ \text{entity}_{n,4} & 0 & 0 & 0 & 0 & 1 \\ \end{array}

What is ( ( ∑ n = 6 ∞ ∑ i = 0 n entity n , i ) m o d 729 ) (\left(\sum _{n=6}^{\infty } \sum _{i=0}^n \text{entity}_{n,i}\right) \bmod 729) ?


The answer is 0.

A Former Brilliant Member by A Former Brilliant Member

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Each column sums to 3 n 3^n , for n ≥ 6 n\geq 6 , the columns modulus 729 are 0. Therefore, the sum is 0 0 .

Note the following (values of ( e + 2 v ) n (e+2v)^n : 0 1 1 1 e + 2 v 3 2 e 2 + 4 v e + 4 v 2 9 3 e 3 + 6 v e 2 + 12 v 2 e + 8 v 3 27 4 e 4 + 8 v e 3 + 24 v 2 e 2 + 32 v 3 e + 16 v 4 81 5 e 5 + 10 v e 4 + 40 v 2 e 3 + 80 v 3 e 2 + 80 v 4 e + 32 v 5 243 6 e 6 + 12 v e 5 + 60 v 2 e 4 + 160 v 3 e 3 + 240 v 4 e 2 + 192 v 5 e + 64 v 6 729 7 e 7 + 14 v e 6 + 84 v 2 e 5 + 280 v 3 e 4 + 560 v 4 e 3 + 672 v 5 e 2 + 448 v 6 e + 128 v 7 2187 8 e 8 + 16 v e 7 + 112 v 2 e 6 + 448 v 3 e 5 + 1120 v 4 e 4 + 1792 v 5 e 3 + 1792 v 6 e 2 + 1024 v 7 e + 256 v 8 6561 \begin{array}{ccr} 0 & 1 & 1 \\ 1 & e+2 v & 3 \\ 2 & e^2+4 v e+4 v^2 & 9 \\ 3 & e^3+6 v e^2+12 v^2 e+8 v^3 & 27 \\ 4 & e^4+8 v e^3+24 v^2 e^2+32 v^3 e+16 v^4 & 81 \\ 5 & e^5+10 v e^4+40 v^2 e^3+80 v^3 e^2+80 v^4 e+32 v^5 & 243 \\ 6 & e^6+12 v e^5+60 v^2 e^4+160 v^3 e^3+240 v^4 e^2+192 v^5 e+64 v^6 & 729 \\ 7 & e^7+14 v e^6+84 v^2 e^5+280 v^3 e^4+560 v^4 e^3+672 v^5 e^2+448 v^6 e+128 v^7 & 2187 \\ 8 & e^8+16 v e^7+112 v^2 e^6+448 v^3 e^5+1120 v^4 e^4+1792 v^5 e^3+1792 v^6 e^2+1024 v^7 e+256 v^8 & 6561 \\ \end{array}

Proof by induction proves the statement.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...