Theoriology

 

“Given a triangle. Is it isosceles?” a teacher asks strictly. “It looks like it,” the student replies.

This is a rather absurd dialogue. Anyone who knows a little about mathematics knows that “looks” don’t prove anything. Two procedures are allowed: a suitable proof and a suitable measurement, or more generally: a suitable observation. What constitutes a “suitable proof” is far from a simple question, but we’ll put that in parentheses for now (trusting we can return to it later).

But what constitutes a suitable observation and measurement? In this area – until now – there has been no real order or law, but we are now beginning to address this gap. There are two major theoretical fields of science where observation plays a serious role: geometry and physics. The role of the observer has become particularly spectacular with the revolution of modern physics. It is as if processes have lost their absolute or objective character, and the way they are observed has become a decisive element. (It’s true that Galileo also dealt with the paradoxes of observing relative motion.) It is a well-known fact that quantum physics has accepted as a law of nature that the foundations of the world behave according to whether someone observes them. Not to mention the true failure of human observation: for thousands of years, people were unable to correctly interpret the structure of the cosmos they observed day and night.

Returning to the important and gentle world of geometry, let’s divide our research area into two parts: simple, “bare-eyed” observations, and constructions (“already known to the ancient Greeks”) with a ruler, a compass, and chalk. It is worth making a mythical figure, a certain (fictional) Theoristos, the common denominator of these two areas, whose task is to observe and construct. The big question is: what (human, superhuman, or even “artificial”) abilities should we give him? What should be allowed and what should be forbidden? This is none other than the science of observation and perception, Teorology.

The rules of geometric construction are well known, but perhaps not sufficiently systematized in a way that an axiomatic theory would require. For example, it must be stated that Theoristos can place his compass on a given point as many times as he wants, that he can always decide whether a point lies on a line or a curve, etc. But a number of more difficult questions arise, e.g., can the compass be placed on a line or curve if there is no marked point on it? An interesting (“multiple-chance”) question is whether placing the compass on a point that was not “given” before marks the point.

An even more difficult question seems to be: can a tangent line be drawn to two circles? It would be particularly important to list the observations we “allow” Theoristos to make, such as drawing a line and marking two points on either side of the line. This quickly reveals the different categories and competencies of observation. This theoretical experiment is an innocent spiritual adventure of the human mind. Today it is difficult to decide whether it promises little or much, but especially now, in the turbulent period of the rapid development of artificial intelligence and machine learning, it can bring about surprises.

 

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S-cycles 1.

 

We can discover interesting relationships if we examine the sequences consisting of the smallest and largest divisors of natural numbers.

By definition, we should not think of trivial divisors here, because then we would get the two most trivial sequences: 1,1,1,1,1... or 1,2,3,4,5,... So the so-called it is worth thinking about your own (or internal) dividers.

Fix the definitions: a natural number a is a non-trivial divisor of a natural number n if and only if 1<a<n and there exists a number b such that a*b=n (in this case it is unnecessary to stipulate separately that b is also a natural number, this necessarily follows from the given conditions).

We would think that, based on the definition, we can now easily assemble the two sequences mentioned at the beginning. But - literally - right away - we are faced with a dilemma: what if a number does not have its own divisor? It seems logical that then the member of the sequences should be 0. (From now on, we will use the OEIS notation method, i.e. the nth member of the sequences will be denoted by a(n).)

In summary: the nth member of the SND sequence is the smallest proper divisor of n, if such exists, or 0, if such does not exist (SND: smallest nontrivial divisors).

On the other hand: the nth member of the LND sequence has the largest proper divisor of n, if such exists, or 0, if such does not exist (LND: largest nontrivial divisors).

Specifically, these sequences look like this:

 

n	SND(n)	LND(n)
1	0	0
2	0	0
3	0	0
4	2	2
5	0	0
6	2	3
7	0	0
8	2	4
9	3	3
10	2	5
11	0	0
12	2	6
13	0	0
14	2	7
15	3	5
16	2	8
17	0	0
18	2	9
19	0	0
20	2	10
21	3	7
22	2	11
23	0	0
24	2	12
25	5	5
26	2	13
27	3	9
28	2	14
29	0	0
30	2	15
31	0	0
32	2	16
33	3	11
34	2	17
35	5	7
36	2	18
37	0	0
38	2	19
39	3	13
40	2	20
41	0	0
42	2	21
43	0	0
44	2	22
45	3	15
46	2	23
47	0	0
48	2	24
49	7	7
50	2	25
51	3	17
52	2	26
53	0	0
54	2	27
55	5	11
56	2	28
57	3	19
58	2	29
59	0	0
60	2	30
61	0	0
62	2	31
63	3	21
64	2	32
65	5	13
66	2	33
67	0	0
68	2	34
69	3	23
70	2	35
71	0	0
72	2	36
73	0	0
74	2	37
75	3	25
76	2	38
77	7	11
78	2	39
79	0	0
80	2	40
81	3	27
82	2	41
83	0	0
84	2	42
85	5	17
86	2	43
87	3	29
88	2	44
89	0	0
90	2	45
91	7	13
92	2	46
93	3	31

 

At first glance, the two sequences are quite similar, but growing differences quickly become apparent. What can be said about them.

First, the coincidence of zeros: If SND(n)=0, then LND=0 and vice versa. This is a trivial consequence of the sequence formula.

It can also be observed that the non-zero terms of SND are always prime (this simply follows from the definitions of "smallest" and "prime"). In contrast, non-zero values of LND are completely mixed. We can simply state that LND(n) is prime if and only if n is the product of two primes. E.g. LND(35)=7, (SND(35)=5). A special case is if n is the square of a prime, then (and only then) SND and LND have the same value (e.g. SND(9)=LND(9)=3).

Another nice, and perhaps useful, but not particularly surprising connection is that if the values are not zero

SND(n)*LND(n)=n

The case can be concluded with this, but it cannot be denied that something small here encourages further thinking.

What can we do, we continue it.

 

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