A Truly Incredible Fact About The Number 37

It was this video from the YouTube channel Veritasium that made me aware of the considerable interest attached to the number 37.

I then found this post from a blogger, Chris Grossack, to be especially helpful in explaining the following:

37 is the median value for the second prime factor of an integer; thus the probability that the second prime factor of an integer chosen at random is smaller than 37 is approximately 50%.

He also uses SageMath for his calculations which was an added bonus. Here is the permalink to the calculation to determine the median using the first 100,000 numbers. The output is shown in Figure 1.

Figure 1

The actual proof is summarised in the information contained in Figure 2 which is more than I can comprehend, but I'll include it here:

Figure 2

The blogger uses the formulae shown in Figure 2 to once again show that 37 is the median value. Here is the permalink and the output is shown in Figure 3.

Figure 3

Of course this is not 37's only claim to fame. Wikipedia has an entry for the number 37 and some of the interesting facts contained in that article include a 3 x 3 magic square with 37 at its centre. See Figure 4.

Figure 4

Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11). I wasn't familiar with the notion of a unique prime and so I'll include a definition from the Wikipedia article here:

A prime \(p\) (where \(p\) ≠ 2, 5 when working in base 10) is called unique if there is no other prime \(q\) such that the period length of the decimal expansion of its reciprocal, 1/\(p\), is equal to the period length of the reciprocal of \(q\), 1/\(q\). For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. The next larger unique prime is 9091 with period 10, though the next larger period is 9 (its prime being 333667). Unique primes were described by Samuel Yates in 1980.

I've written about 37 extensively as well in a post titled Star Numbers from the 7th of June 2019. There is a website dedicated to the number 37. It's mentioned by its creator in the YouTube video earlier but, as he himself admits, it hasn't been updated in very many years. However, it still contains a wealth of information.

For example, the site describes a method of determining if a number is divisible by 37. This is the method:

• Divide the number up in groups of three digits, starting from the right.
  (The left-most group may not have three digits.)
• Add the groups together.
• Repeat steps 1 and 2 if the result is still longer than three digits, repeat steps 1 and 2.
• Examine the final three-digit (or smaller) number

The original number is divisible by 37 if and only if this three-digit number is.

I often take note of car number plates here in Jakarta. These are typically of the form B-xxxx where xxxx is a four digit number. It's easy to determine if the four digit number is divisible by 3 because the first digit is simply added to the remaining three. Using leading zeros, the multiples of 37 are:

037, 074, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 703, 740, 777, 814, 851, 888, 925, 962, 999

The repeated digit numbers (111 to 999) are a dead given away but the others are not two difficult to identify. Let's consider a number plate like B-1258. The 1258 --> 1 + 258 = 259 = 7 x 37. In this case, the 7 can be divided in to reveal the 37 rather than dealing with division by 37. There is no limit to what can be said about the number 37 but at least in this post and my earlier post of star numbers I've made a start.