Semiprime Factor Ratios

All biprimes (or semiprimes or 2-almost-primes) can be visualised as unique rectangles and all triprimes (or 3-almost-primes) as rectangular prisms. I only intend to deal with biprimes in this post. Let's take a recent biprime, 24581 = 47 x 523, as a starting point. It can be visualised as a rectangle with a width of 47 units and a length of 523 units. It's the ratio of width to length that's of interest. 

A golden semiprime is defined as a number that factors to: 

• \(p \times q\) (with \(p<q\)) and
• \( |p \times \phi-q|<1 \), where \( \phi\) is the golden ratio of \( \dfrac{1+\sqrt 5}{2} \)

Clearly 24581 does not satisfy this condition and not many semiprimes do. The next for me is 27641 which factors to 131 × 211 and where:$$|131\times \phi-211| \approx 0.9624525$$and so it just barely satisfies the criterion. Here is a partial list as shown in OEIS A108540:

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423

Presumably there is an infinity of golden semiprimes. There are other ratios of interest, for example pi. Here the number 154 = 7 x 22 could be treated in a manner similar to the golden semiprimes and the question asked as to whether \( |7 \times \pi-22| \) is less than 1. It turns out that it is (0.9911...) and so could perhaps be termed a circular semiprime. The number 15883 = 71 x 223 yields a much closer result (0.053...). Similarly for \(e\), the number 133 = 7 x 19 yields \( |7 \times e - 19| \approx 0.02797 \) and could be termed an Euler semiprime for want of a better term. 

Some semiprimes are not related to special mathematical constants but are nonetheless of interest. For instance, for Friday 26th August 2016 (the day I'm completing this post), my number 24617 = 239 x 103 and the ratio 239:103 can be expressed approximately as 2.32:1 (rounding off 2.320388... to two decimal places). This is very close to the aspect ratio for the current widescreen cinema standard of 2.35:1 or 2.39:1. However, following the pattern for the golden semiprime ratio, the result of \( |103 \times 2.35-239|=3.05 \) and \( |103 \times 2.39-239|=7.17 \) mean that the results are outside the acceptable range (less than 1).

Another way to view the ratio 239:103 is as 0.69883:0.30117 and if we round off to two decimal places, the result is 0.70:0.30 or 70% : 30%. This is the ratio of copper to zinc in so-called Cartridge brass described as follows:

70/30 brass has excellent ductility and good strength. It is often used where its deep drawing qualities are needed. The alloy is the most common brass in sheet form (source).

I guess the concept of the golden semiprime has opened my eyes to other classifications of semiprimes based on other constants such \(e\) and \( \pi\). Expressing the ratio in such a way that both sides sum to 1 is also useful because, as in the case of 0.70:0.30, connections to physical applications can be drawn.

on August 30th 2021