Binary Display

I noticed that Numbers Aplenty had the display shown in Figure 1 as part of its output for the number associated with my diurnal age today (28337) with the comment that "when arranged in a rectangular grid, its binary digits exhibit some kind of symmetry". The binary digits are 110111010110001 and there are 15 of them so they can be arranged in a 5 x 3 grid.

Figure 1

I quite liked this output and decided to incorporate it into the program for my daily number analysis. The output is as shown in Figure 2.

Figure 2

Every day, from this day forwards, there will be a similar display and I can decide whether there is anything of interest regarding the display, symmetrical or otherwise. Figure 3 shows the results for 27334 where there is also lateral symmetry.

Figure 3

The range of numbers for which there are 15 binary digits is 16,384 to 32,767 corresponding to \(2^{14}\) and \(2^{15}-1\). Prior to 16384, there is a range of numbers with 14 binary digits that can also be arranged in a rectangular grid. This range is from 8192 to 16383 corresponding to \(2^{13}\) and \(2^{14}-1\). Figure 4 shows the display for 16382:

Figure 4

Above 32767, we enter the realm of 16 binary digits and no grid will be displayed. The output for 32768 would be:

Integer : 32768
Binary  : 1000000000000000
Length  : 16 binary digits

Result  : Bit length 16 (factorization: 2^4) is NOT a biprime. 

No grid generated.

Of course, a 4 x 4 or 8 x 2 grid could be generated with a tweak of the algorithm but I'll consider that as I approach 32768 by which point I'll be 89 years old.