23501 and the Visualisation of Biprimes

Today I'm 23501 days old. This number is biprime, meaning that it has two prime factors (71 and 331). One way to get a handle on understanding the significance of biprimes is to imagine that you have an area of 23501 to enclose with a rectangular fence made up of modular 1 metre sections. This means of course that both sides of the rectangle must be whole numbers. There is only one way to do this and that is by creating a rectangle with a length of 331 metres and a width of 71 metres.

With other numbers that are neither prime nor biprime (such as 23502), there is more than one way. For example, because 23502 = 2 x 3 x 3917, there are three possible rectangles:

• 2 metres by 11751 metres representing a perimeter of 23.506 kilometres
• 3 metres by 7834 metres representing a perimeter of 15.674 kilometres
• 6 metres by 3917 metres representing a perimeter of 7.846 kilometres

These are admittedly very narrow rectangles but rectangles none the less. Of course, they are very inefficient in terms of the perimeter length required to cover the required area. Without the restraint of whole numbers, the most efficient four sided figure would be a square with a side of a little more than 153.3 metres and perimeter of about 613.2 metres or 0.6132 kilometres.

You could measure, in percentage terms, the efficiency of the perimeter required to enclose the area by using the formula:$$ \text{efficiency}=\frac{ \text{perimeter of square}}{ \text{perimeter of rectangle}} \times 100$$In this case, the efficiencies become:

• 2.6%
• 3.9%
• 7.8%

Getting back to 23501 however, the efficiency of the rectangle turns out to be 76.3%. Of course, with more factors the number of possible rectangles increase and rectangles of greater "perimeter efficiency" can be defined. Of course, biprimes with equal factors will convert to squares and thus have 100% efficiency.

Reflections on 23500

Today I'm 23500 days old and I was looking around to see if there was any significance to this number. It's factors are unremarkable (2^2 x 5^3 x 47) but the fact that it's halfway between 23000 and 24000 means that it pops up quite frequently in Internet searches, as would 22500 or 24500 I would imagine. A search reveals that the approximate population of Boston in 1620 was 23500 and there are several towns around the world that are listed as having this population currently e.g. Bishopbriggs in Scotland.

Bishopbriggs grew from a small rural village on the old road from Glasgow to Kirkintilloch and Stirling during the 19th century, eventually growing to incorporate the adjacent villages of Auchinairn, Cadder, Jellyhill and Mavis Valley. It currently has a population of approximately 23,500 people.

Source: http://en.wikipedia.org/wiki/Bishopbriggs

It turns out that Mount Isa has the same population (source):

Mount Isa is located just 200 kilometres from the Northern Territory border and 1,829 kilometres from Queensland’s capital, Brisbane. The nearest major city, Townsville, can be found 883 kilometres from The Isa. Mount Isa covers an area of over 43,310 square kilometres, making it geographically the second largest city in Australia to Kalgoorlie-Boulder, Western Australia ... With a population of approximately 23,500, Mount Isa is a major service centre for north-west Queensland.

Many other examples of towns having populations of about 23500 could be quoted. In addition of populations, the number sometimes comes up as a dollar figure (source):

In its third annual funding cycle, the Black Philanthropy Initiative has pumped $23,500 back into the Winston-Salem area to help African Americans improve their parenting skills.

Interestingly, it turns out that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source).

The sun is far enough away (about 23,500 earth radii) that it took a long time before people knew accurately how far away the sun was. Certainly the ancient Greeks had calculated the distance, but they also knew that their results could be off. 

Many countries in the world have five digit postal codes or zip codes as they are sometimes known. These codes identify particular locations within the country e.g. Muang Prachinburi, Prachinburi, Thailand has a postcode of 23500. The United States uses a five digit system but apparently there is no location corresponding to 23500, although there is for 23499 and 23501.