Visualisation of Semiprimes

I've written previously about semiprimes in varying contexts. These posts are listed below:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• 2019: A Numerical Profile
• Semiprime Factor Ratios
• More about Golden Semiprimes

I was prompted to make yet another post about them because today I turned 25661 days old and was little of interest to found about this number in either the OEIS, Numbers Aplenty or any other sources. However, it is a semiprime, being a product of 67 and 383. I thought I'd examine the number is a more detailed, two dimensional way. Figure 1 illustrates my approach.

Figure 1

I've revisited some old territory in the notes contained in Figure 1 which I've reproduced below:

The number 25661 is a semiprime because it has prime factors of 67 and 383. It can be visualised in two dimensions as a rectangle with a width of 67 units and length of 383 units. As such, its area of course is 25661 square units and its perimeter is 900 units. The ratio of the rectangle's width to its length is thus 67 383 or approximately 0.17493 and the ratio of length to width is 383: 67 or approximately 5.7164. The length of the diagonal of this rectangle is approximately equal to 388.8. The area of the rectangle is equivalent to the sum of the areas of 62 different combinations of three squares. An example is shown where the three squares has sides of 86 units, 92 units and 99 units.

By "old territory", I mean the semiprime is envisioned as a rectangle whose width and length are the smaller and larger prime factors whose product is the area of the rectangle. Thus the integers 900 and 25661 are related via a gematria-like connection. The ratio of the sides produce two other related numbers, both rational, and this case:$$ \frac{67}{383} \approx 0.17493 \text{ and its reciprocal } \frac{383}{67} \approx 5.7164$$What's new is that I've considered the length of the rectangle's diagonal which is an irrational number and equal to \( \sqrt {67^2+383^2} = \sqrt {151178} \approx 388.8162 \).

Finally, I've used the fact that 25661 can be expressed a sum of three squares in 62 different ways to represent the rectangle as being equivalent in area to the sum of any of these three squares. In Figure 1, I've used the example of:$$86^2+92^2+99^2=67 \times 383 =25661$$The square numbers correspond to \( 86^2, 92^2 \text{ and } 99^2 \text{ are } 7396, 8464 \text{ and } 9801 \text{ respectively }\). There are another 61 sets of such triplets that can be linked visually with the rectangular representation of 25661. See Figure 2:

Figure 2