4.03.2021

A completive sequence

 

The sequence:

 

1, 1, 5, 7, 4, 6, 3, 5, 2, 4, 1, 3, 9, 2, 8, 1, 7, 11, 6, 10, 5, 9, 4, 8, 3, 7, 2, 6, 1, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 15, 8, 14, 7, 13, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 17, 6, 16, 5, 15, 4, 14, 3, 13, 2, 12, 1, 11, … (A343039)

  

The definition of the sequence:

a(1)=1, for n>1, a(n) is the smallest positive integer for which a(n-1) + n + a(n) is  square.


Hypothesis: the sequence has an infinite number of 1 members.

 

Other similar sequences can be easily created by writing a different number type instead of a square in the definition (prime number, Fibonacci number, etc.).

 



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12.19.2020

Gentle dynamics

 

It is well known that dynamic systems are typically unstoppable, so, as Poincaré has already shown, there is some mysterious obstacle in the way of any long-term forecast.

A dynamic system can consist of many, but it can even consist of a variable, in which case we can speak of a simple sequence. But what makes a system or a sequence dynamic. Basically, because it is a function of time, namely discrete, it is a function of time consisting of some unit of time. This will really be the function sequence.

But dynamic systems have another important feature: the new values ​​depend on the “past”, that is, not only on the value of the variable, but on the values ​​taken in previous places in the function - in the case of a sequence, on the preceding members.

One of the best known dynamic sequence is Fibonacci numbers. This is not really a family of one, but an infinite number of sequence with very similar properties. These are truly dizzyingly exciting, thoroughly researched sequence, and by the way not so extreme, yet even in the gentlest version, today’s Excel around the 50th member “throws in the towel”.

Meanwhile, it must be acknowledged that it is difficult to imagine a more gentle than the Fibonacci formula. Anyone trying to compose some kind of dynamic sequence faces a similar difficulty: values ​​grow at a dizzying pace over time, even if some cyclicality develops in the meantime. Therefore, it was a pleasant surprise to find an absolutely gentle dynamic sequence that also works with the square of values.

Here's the sequence:

 

1, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, …

 

No doubt, it is really gentle and in a very nice dance move, a real szirtaki! What formula could he have born?

Here is the formula for the sequence:

 

a(n) = a(n-2)^2 + a(n-1)^2 – 2*a(n-2)*a(n-1) – a(n-2) – a(n-1) +n + 1

 

You can try with other starting values and you will immediately experience the aforementioned difficulties of prognosis.

But why this strange three-step cycle?

 



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10.23.2020

Primes - four conjectures

 

We consider prime numbers, rightly, as elementary particles of all numbers. Meanwhile, the situation is that we know (almost) everything about numbers, very precisely, but we have only “empirical” knowledge and various hypotheses about prime numbers themselves that can only be obtained by calculations. It is as if something is blocking the paths of omnipotent theoretical research to the mystical primes. Although this is another topic (which you will want to return to.)

We would now consider four related hypotheses giving to this some “empirical” statements.

A well-known concept is the twin prime. Here, care must be taken that the term itself can be applied to both a number and a pair of numbers, and this requires attention. So, an n number is a twin prime if prime and either n-2 or n+2 is prime also. That is, there is another prime “near” n. Two numbers n and m are twin primes or twin prime pairs if both are prime and n-m=2 or m-n=2. The topic and wordings can be complicated by the strange beginning of the primes (2, 3, 5, 7). This “weird start” is also a separate topic (which is also worth returning to). Now we can do this by counting the twin prime numbers from 5 onwards. And indeed, the first “regular” twin pairs: (5,7), (11,13), (17,19), and so on.

Well, these twin primes are of huge interest, many deal with them, you can read a lot about them. But what are non-twin prime numbers like? Many times, if there is a highlighted subset, there may be several others besides it, but this is not the case here (but like the concept of an even number: if a number is not even, it is odd, and that’s it). Indeed, if there is no other prime “near” a prime number, then there is not, and that is it. It seems legitimate to call such primes alone or lonely. That is, not every prime or twin, or lonely, third case. The first reflexive question in arithmetic is that there are an infinite number of these? Well, let's not wait for a theoretical, demonstrable result. However, we can have a hypothesis.

 

H1 (Small twin prime hypothesis) There are infinitely many twin primes.

 

H2 (Small one-prime hypothesis) There are infinitely many one-primes.

 

The first hypothesis has been known for a long time, it must have been formulated when we first talked about twin primes. All that can be added is that many have tried to prove it, and often partial results are reported (a partial result of the siege of an old girl may also be that she is already half-virgin, but here we are still far from it). The question, however, is really interesting because the statement can by no means be said to be obvious.

In contrast, questioning Hypothesis H2 would be pure absurdity. Anyone can rightly say: this statement is infinitely trivial. Indeed. At a glance. But let me prove it! I find the proof of this statement as hopeless as that of H1, and that is why it deserves the status of a written hypothesis. In the meantime, I do not rule out that this was first described in this form.

Now let’s move on to examining these two animal species.

An obvious and legitimate question is: if these two kinds of prime numbers va, and presumably there are infinitely many of both, how are they distributed? It immediately becomes apparent that the twins are starting strongly: the six primes starting with 5 are all twins, but they are also in good numbers when viewed up to 100. First, however, let us note that, except for the departure curiosity, where 3, 5, and 7 are no three consecutive twins, triple twins, which is - easily - provable.

 

Lemma 1. If p > 3 and p and p + 2 are prime, p + 4 is not prime.

 

As a result, you need to “rest” after each pair of twin primes. All right, come the rest, but the rest alone does not decide whether it is followed by a twin or a single. For example, after 7 there is a twin, after 19 there is single. There is nothing surprising in this that twin prime pairs can form groups. It is not too difficult to map these. This way we can create sequences that identify (with their first member) the different groups. A more difficult question is how large these groups can be. It is quite understandable that the larger such groups we search for, the less often we find such. And here the difficulties arise surprisingly quickly. Even the discovery of groups of 5 pairs of twin primes requires effective computer assistance. Currently, the research competition is stuck in groups of 10. Sure, we cannot doubt that our ever-growing computers are making further progress, discovering groups 11 and 12, but the members of these are already inconceivable numbers.

Yet, the human mind does not reach that much. The question is: for any n numbers, is there a group containing n pairs of twin primes? Let's not expect a theoretical result for this either, but let us also have the courage to formulate the third hypothesis:

 

H3 (Big Twin Hypothesis) For any natural n number, there exists an n-member group of prime pairs.

 

Before we go any further, let’s take a look at some of the series related here

Starting numbers of twin prime pairs:

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607 (A001359)

 

Starting numbers for twin-pair groups:

5, 11, 101, 137, 179, 191, 419, 809, 821, 1019, 1049, 1481, 1871, 1931, 2081, 2111, 2969, 3251, 3359, 3371, 3461, 4217, 4229, 4259, 5009, 5651, 5867, 6689, 6761, 6779, 6947, 7331, 7547 (A053778)

 

Summary - Starting numbers for n-member twin-pair groups:

3, 5, 5, 9419, 909287, 325267931, 678771479, 1107819732821, 170669145704411, 3324648277099157 (A111950, In 2011, Gábor Lévai announced that he had found a group of 11)

 

Now let’s turn our attention to the somewhat marginalized single primes. First, let's say the fourth promised hypothesis:

 

H4 (Big one-prime hypothesis) For any n natural numbers there exists a n-member one-prime group.

 

Undoubtedly, this is an “obvious” statement like H2, but here it can be said that it does not seem to be formally provable. Nevertheless, here we can make calculations similar to twin primes, which are somewhat less common in the literature for single primes.

 

The single primes:

23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563 (A007510, although this - in my humble opinion - incorrectly includes 2 as well)

 

Starting number of double single prime groups:

47, 79, 83, 89, 113, 127, 157, 163, 167, 211, 251, 257, 293, 317, 331, 353, 359, 367, 373, 379, 383, 389, 397, 401, 439, 443, 449, 467, 479, 487, 491, 499, 503, 541, 547, 557, 577, 587, 607, 647 (A126095)

 

And for the summary of single-prime groups, i.e., the starting number of n-membered single-prime groups:

23, 47, 79, 79, 353, 353, 353, 353, 353, 673, 673, 673, 673, 673, 673, 673, 673, 8641

 

Before concluding on this topic, I would like to point out a useful fact. To do this, define two concepts: if p is a prime number, we call foreground the difference between p and the previous prime, the background is the differences between the following prime and p, denoting them FG(p), resp. BG(p). Example: , FG(13) = 2 and BG(13) = 4

Obviously:

 

Lemma 2. For any prime p greater than 3, FG(p) and BG(p) are even numbers.

 

The following is less trivial, but very useful:

 

Lemma 3. For any prime p, if , FG(p) = BG(p), then this number is 6 or some multiple of it.

 

The distribution of the two types of primes is best shown in the following series, where the numbers are the “isolation index” of the primes, which is the minimum of FG(p) -1 and BG(p) -1, i.e. the smaller of the number of composites before and the number of the composites after p. It follows that (with the exception of 3) a prime is a twin prime if and only if it has an isolation index of 1. Otherwise, it is a solitary prime, which is more solitary the higher its isolation index.

 

0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 1, 1, 3, 3, 5, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 5, 3, 3, 5, 1, 1, 1, 1, 1, 1, 11, 3, 1, 1, 3, 1, 1, 5, 5, 5, 1, 1, 3, 1, 1, 9, 3, 1, 1, 3, 5, 5, 1, 1, 3, 5, 5, 5, 3, 3, 5, 3, 3, 7, 1, 1, 1, 1, 3, 3, 5, 3, 1, 1, 3, 7, 3, 3, 3, 3, 5, 1, 1, 5, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 5, 3, 1, 1, 9, 1, 1, 3, 5, 1, 1, 3, 3, 5, 7, 7, 7, 7, 5, 5, 3, 3, 5, 3, 3, 3, 3, 9, 9, 1, 1, 1, 1, 1, 1, 9, 3, 1, 1, 3, 3, 1, 1, 3, 3, 3, 7, 7, 3, 3, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 1, 3, 5, 5, 5, 5, 5, 1, 1, 7, 7, 5, 5, 5, 7, 3, 3, 5,

 



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10.08.2020

Diophantine rectangles

 For any n, if n squares (tiles) are given, we can lay out a rectangle in them. If n is a prime number, then the only rectangle that can be unloaded is only formally rectangle, in fact, a row.

Meanwhile, it is true that for any of n, this quantity of squares can always be unloaded in a row. Therefore, and because of the usual aesthetic requirements, we could try to lay out a rectangle that most closely resembles a square. It is clear that a square can be laid out if and only if n is itself a square number.

If n is neither a prime nor a square, we can lay out as many different rectangles as the number of proper divisers of n divided by 2 (this number does not include the case of squares placed in a row). It is easy to decide which rectangle is considered to be closest to the square.

A special case if n is a so-called semiprime (i.e., the product of two prime numbers). Then we can unload a single “prime rectangle” from this.

An interesting question is when we can unload another prime rectangle by adding 2 squares to the original quantity.

Several similar, very interesting questions can be raised on this topic. For example, if the height of the desired rectangle is specified in advance, which is the smallest width?

For such questions, a solution formula cannot usually be given, but the results can be well calculated in function of n, in the head or with a program.

We will provide several examples of these soon. But we are also happy to share the results with others.

 



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7.12.2020

System’s question

What about

1872870801843041394471000000000

Nothing special, this is the number that is in the Simi numeral system

99999

 

Siminumeral system

 


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