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.