37 ३७

It is often heard that 37 is the most common choice when people are asked to choose a 2-digit random number, but let me clearly mention that this article does not contain the faintest flavor of numerology #数秘 though it is full of numbers #数字 .

In my previous article I wrote "...if you ask me what my favorite numbers are, my answer would be 71, 907, 1093, 433494437, 7758337633 (all prime numbers) et al. for integers and Ramanujan's constant and the first Feigenbaum constant δ to the period-doubling bifurcations in the logistic map, if not restricted to integers."

However, 37 did not show up. Since I described my favorite numbers, which are NOT randomly selected for an obvious reason.

Relative to 37 being a surprisingly common choice upon a request for random pickup of a 2-digit number, there are many articles, and also it appears that random number surveys were actually conducted. However, It is not my intention to find commonalities among those articles. In addition, if there's any psychological aspect in a process of choosing a number, it is not my interest either to pursue whatever it is.

I found the following comment though:

"Another theory is that the number 37 has some inherent mathematical properties that make it useful in certain programming contexts."

So let me share some interesting mathematical properties of 37😉

• The only prime with period length three: 1/37 = 0.027 027 027 ....

• The smallest irregular prime, where regularity/irregularity of prime is defined as follows:

• The class number of cyclotomic field for p= 37 is 37. This is a big jump from 3 (for p=23), 8 (for p=29) and 9 (for p=31).

• Appears in values of Riemann ζ Function (for negative odd integers) and in Bernoulli numbers:

• As for expression by sum of 3 cubes, 37 is an extremely easy case: 37 = 0^3 + (-3)^3 + 4^3

[NOTE 1] This particular simple relation 64 - 27 = 37 leads to the following example 3.2 in 8.3. Fibered Surfaces, Chapter 8. Birational geometry of surfaces of "Algebraic Geometry and Arithmetic Curves" (by Qing Liu):

This has nothing to do with Boeing X-37, a.k.a. the Orbital Test Vehicle (OTV), which appears in the header of this article.
(Image via Wikipedia)

[NOTE 2] 33 and 42 are notable examples in this "sums of three cubes problem". Here is an article by Quanta Magazine:

Aside from its mathematical properties, 37 appears in real world problems.
Suppose you encounter a problem where you cannot assess all the options at once and you have to decide with each option you encounter. Then you need to decide whether to accept it or reject it forever and see what comes next.

In these scenarios, one would feel it's impossible to make the best choice - either a premature selection or ending up with a negative selection.
So your best bet is somewhere in the middle.

This is known as the Secretary problem, and an excerpt from Wikipedia is the following:

...It implies that the optimal win probability is always at least 1/e (where e is the base of the natural logarithm)...

A calculation of the probability would lead to the following (depicted using Desmos), and this is how 37% comes from. 

For those who find the name "Secretary problem" inappropriate in the 21st century, please be advised that Wolfram calls it "Sultan's Dowry Problem" with the following page😎:

37 has some other properties from recreational mathematics point of view:

• Factorization of numbers of the form 10^n + 37:

• On the other hand, numbers of the form 37 * (10^n) + 1 are prime for the following n: 2, 5, 12, 17, 120…

n=2 : 3701
n=5 : 3700001
n=12 : 37000000000001
n=17 : 3700000000000000001
...
n=120: 37000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 (122 digits)

• 2^3 + 5^7 + 11^13 + 17^19 + 23^29 + 31^37 = 15148954872646850196557152427604893685308877022260348791 is prime (56 digits).

• 2^37 + 3^31 + 5^29 + 7^23 + 11^19 + 13^17 + 17^13 + 19^11 + 23^7 + 29^5 + 31^3 + 37^2 = 283453407513524913023 is prime.

• The prime p = 37, and its reversal q = 73, are the only known emirp pair such that p! + 1 and q! + 1 are both primes.

• 2! + 3! + 5! + ... + 29! = 31 (mod 37). It is the largest known prime with this property.

• The sum of the first 37 primes is a Fibonacci number, and please note 1597 is a Fibonacci prime.

One could spend entire life to come up endless examples…but I'm under the impression that one would not find a gem without theories… 😶