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?

 

* * *

 

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,

 

* * *

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.

 

* * *

System’s question

What about

1872870801843041394471000000000

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

99999

 

Siminumeral system

 

* * *

Aristarchus sequences

While writing my book The Numbers, I came up with dozens of more or less interesting sequences. I wrote some for myself, not others, but in each case I had the intention of returning to them “almost once”.

So in the days I took out one and discovered some very interesting qualities in him and - to my delight - all the more mystery.

Since it’s a whole family of sequences, I felt it would be nice to give them a name. So I started calling myself the Aristarchus sequences. He who accepts this designation accepts who does not. In any case, Aristarchus deserves our respect.

But let's move on!

The basic sequences is formed according to the following rule:

a(1) = 1, if n> 1, a(n) = a(n-1)/gcd(a (n-1), n) if gcd(a(n-1), n) > 1, if gcd(a(n-1), n) = 1, then a(n) = a(n-1) + n.

In words, if the largest common divisor of the previous member and n is greater than 1) so is a real divisor), then we divide the previous member by that, and this becomes the new member, but if the previous member and n are relative prime, the new member is the sum of the previous member and n.

There are no special “screws” in the rule itself, we can wait with interest for the result.

Well, that might even seem disappointing, because this sequences looks like this:

1, 3, 1, 5, 1, 7, 1, 9, 1, ….

That is, it is fairly unanimous and predictable.

If that were all, we might even forget the sequences.

But let’s see what happens if the first member is not 1, but - for the sake of order - 2!

A little surprise. The sequences now looks like this:

2, 1, 4, 1, 6, 1, 8, 1, …

Exemplary order and discipline, almost beautiful! While every sequences would be like this, we would have no problem with them.

But the mathematician is curious, let’s see how this sequences looks like for other novice members when we talk about it so much!

Let's see what the sequences is like if the first member is 3!

Well, you should hold on now! The sequences looks like this:

3, 5, 8, 2, 7, 13, 20, 5, 14, 7, 18, 3, 16, 8, 23, 39, 56, 28, 47, 67, 88, 4, 27, 9, 34, 17, 44, 11, 40, 4, 35, 67, 100, 50, 10, 5, 42, 21, 7, 47, 88, 44, 87, 131, 176, 88, 135, 45, 94, 47, 98, 49, 102, 17, 72, 9, 3, 61, 120, 2, 63, 125, 188, 47, 112, 56, 123, 191, 260, 26, 97, 169, 242, 121, 196, 49, 7, 85, 164, 41, 122, 61, 144, 12, 97, 183, 61, 149, 238, 119, 17, 109, 202, 101, 196, 49, 146, 73, 172, 43, 144, 24, 127, 231, 11, 117, 224, 56, 165, 3, …

Undoubtedly, this looks staggering after the first two cases. You could say a real random sequence. Indeed, what connections can be discovered here? (Warning: this is just a provocative question.)

And now may come the real surprise! Right here, where I left off, this bizarre unruly sequence returns to the well-known chiseled bed and continues like this:

1, 113, 1, 115, 1, 117, 1, 119, 1, …

The case is, to put it mildly, interesting, and once discovered, an honest numerator does not leave it at that, but examines it with all natural numbers. (Not big so, this is the smallest infinite set.)

What a denial, this is also an interesting sequence when the sequences returns to normal.

Here are the modest first results (here I will only give the results that form a continuous line):

1, 2, 111, 7, 5, 3, 25, 22, 25, 111, 111, 4, 7, 5, 5, 6, 22, …

Why does 111 occur so many times, why does it occur twice in a row (if the sequences starts with 10 or 11)?

Dozens of questions, dozens of mysteries. We will certainly never get an answer to some of them, yes to others. And I'm looking forward to them.

Finally, another important additive to liven up life in the world of the Aristarchus sequences. Well, a small change in the basic formula dramatically changes its character, leaving no trace of this old monotonous bed. And this change is that if gcd is 1, then the new term is the sum of the previous and n, minus 1. Then the sequence looks like this:

1, 2, 4, 1, 5, 10, 16, 2, 10, 1, 11, 22, 34, 17, 31, 46, 62, 31, 49, 68, 88, 4, 26, 13, 37, 62, 88, 22, 50, 5, 35, 66, 2, 1, 35, 70, 106, 53, 91, 130, 170, 85, 127, 170, 34, 17, 63, 21, 3, 52, 102, 51, 103, 156, 210, 15, 5, 62, 120, 2, 62, 1, 63, 126, 190, 95, 161, 228, 76, 38, 108, 3, 75, 148, 222, 111, 187, 264, 342, 171, 19, …

We see here that the appearance of 1 does not lead the sequences anywhere, boredom is ruled out.

That is now the first news of the Aristarchus sequences.

* * *

 

Self-powered numbers

We recently asked the not-so-complicated question: why is the

387 420 489

number interesting?

The answer is: because it is the largest number that can be written with two digits:

387 420 489 = 9⁹

There are people who, if they are not in a good mood, do not see anything interesting in anything. They can be heartily regretted. Let’s keep up our good humor and try now to see something more interesting in this issue!

Well, I’m not sparking interest: the amazing thing is that 9⁹ is properly written out of nine digits. Could this be typical of nⁿ type numbers? What a pleasant start: 1¹ is a single digit. Capture us with excitement…

But the series doesn’t go on like that, it’s stuttering locally. 2² also consists of one digit. This disadvantage can no longer be worked out – until 8. 8⁸ consists of 8 digits.

For 10¹⁰, 10 zeros is enough, but it still need 11 digits. From now on, however, there is no stopping. Above this, to describe nⁿ, it would need an increasing number of digits that n. However, this sequence does not give a dizzying progression. Monotonously growing but gently growing. We do not give examples, but show the first 80 values. Interesting sequence. Unless you're in the mood.

* * *

Friendly neighbors

There are no two more different numbers than two adjacent numbers (i.e., than n and n+1, to make us feel: we are in the field of mathematics). If one number is divisible by a third, it is certain that its neighbor is not. And vice versa. Perfect antipodes.

Is that wrong? What do the properties of numbers mean in the end? Earth is n kilometers from the Sun, not long after that distance is n + 1. Everyone may ask: and then what has changed?

Well, that adds to the greatness of mathematics in that it does not deal with such unnecessary questions, but unbrokenly examines everything that can be found in the field of the properties of numbers.

Therefore, we are also now looking at the question: is there still nothing that connects at least some neighboring numbers, such as a common hobby connecting two neighbors in the city?

Modest merit, but still something if the sum of two neighboring numbers is prime. After all, this is not the case for all consecutive numbers. For example, the sum of 5 and 6 is prime (11), but 7 and 8 is not (15). We can call such numbers P-friendly neighbors. Nothing is easier to define:: all primes (and by implication only those) clearly define such a P-friendly pair of numbers. Otherwise, the property may be somewhat reminiscent of the Goldbach conjecture (one states of even numbers and the other of prime numbers that it can be written as the sum of certain two numbers). The difference is that the first is a presumably unprovable conjecture, but in innumerable true cases it can be solved in several ways, and the second is an easily provable theorem, which in turn has a single solution in specific cases.

A somewhat similar property is when the sum of two adjacent numbers is a square number. Then we can say that the two numbers are Q-friendly numbers. Here it is easy to notice that the sum of two adjacent numbers can never give an even square number, and we can boldly generalize this serious observation: any sum of two adjacent numbers is odd, which can be reversed: every odd number can be written - in a clear way - as the sum of two adjacent numbers.

Starting from the sum of the neighbors, we can endlessly define different similar properties, but now let’s look at a more exciting option. As mentioned, all two neighboring numbers are totally different with respect to their divisors. But is it possible that their total number of divisors is the same (they could be D-friends). The answer is yes, and it is not hard to get stuck quickly first, including the first non-trivial examples. After all, in such a “trivial” way, 2 and 3 are already D-friends.

Returning here to the even-odd question: it is easy to notice that the P- and D-friendly numbers are even and odd, and presumably no substantial regularity can be discovered in this respect. In contrast, Q-friendly numbers are all divisible by 4 (but not always by 8).

We will continue the topic. We would also be happy to share the results of others in this regard.

Finally, we give the first hundred numbers corresponding to the above three properties

P-friendly numbers: 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 173, 174, 176, 179, 183, 186, 189, 191, 194, 198, 200, 204, 209, 210, 215, 216, 219, 221, 224, 228, 230, 231, 233, 239, 243, 245, 249, 251, 254, 260, 261, 270, 273

Q-friendly numbers: 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324, 4512, 4704, 4900, 5100, 5304, 5512, 5724, 5940, 6160, 6384, 6612, 6844, 7080, 7320, 7564, 7812, 8064, 8320, 8580, 8844, 9112, 9384, 9660, 9940, 10224, 10512, 10804, 11100, 11400, 11704, 12012, 12324, 12640, 12960, 13284, 13612, 13944, 14280, 14620, 14964, 15312, 15664, 16020, 16380, 16744, 17112, 17484, 17860, 18240, 18624, 19012, 19404, 19800, 20200

D-friendly numbers: 1, 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 104, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 189, 201, 202, 205, 213, 214, 217, 218, 230, 231, 242, 243, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 344, 374, 375, 381, 387, 393, 394, 429, 434, 445, 446, 453, 481, 501, 507, 514, 526, 537, 542, 548, 553, 565, 603, 604, 609, 622, 633, 634, 645, 663, 664, 694, 697, 698, 706, 717, 724, 735, 741, 745, 766, 776, 778, 782, 793, 802, 805, 817, 819

* * *

Question – This number wit fewer difitd?

* * *

Alone with Numbers

* * *

New family of Fibonacci Sequences 1

The Fibonacci sequences has a long and widely known reputation. It grabs your attention for a reason, as it has tremendously exciting features. So it is not surprising that he also inspired various generalizations. There is a fascinating literature on the subject, and anyone can easily believe that here we cannot expect big news.

Such a statement would be very careless. We should know that mathematics will not run out, and certainly will never run out of surprises.

For some time now I have been dealing with some issues in the sequences, and in the meantime I came across a little surprise.

At first, looking at a special version of the Fibonacci sequences, I was struck by a dramatic change. Well, now is the time to revive the definition of the Fibonacci sequences for the sake of order and to record some notations.

The Fibonacci sequence is the sequence of numbers denoted by F(n) in natural numbers such that F(1) = 1, F(2) = 1, and (if n> 2) F(n) = F(n -2) + F(n-1). The essence of the definition is that the first two terms of the sequences are given separately and in advance, from then on each subsequent member is the sum of the previous two. Since this is a summation of numbers (not, for example, division), we cannot expect any "difficulty" in calculating the sequences.

The Fibonacci sequence has a long and widely known reputation. It grabs your attention for a reason, as it has tremendously exciting features. So it is not surprising that he also inspired various generalizations. There is a fascinating literature on the subject, and anyone can easily believe that here we cannot expect big news.

Such a statement would be very careless. We should know that mathematics will not run out, and certainly will never run out of surprises.
For some time now I have been dealing with some issues in the sequences, and in the meantime I came across a little surprise.

Immediately, the first two terms are very arbitrary, and this is where most of the generalization of the sequences comes from. That is, any sequences with a first member a and a second member b is considered a Fibonacci sequences (the other rule remains the same for the other members). This means that any pair of numbers (a,b) can be assigned to a distinct sequence, which is obviously denoted by F_a,b(n).

It is very easy to see some striking features, such as selecting two consecutive members from a given sequences and launching a new sequences with them is not really new at all, on the contrary the new sequences is completely identical to the original from the two selected numbers.

Among other things, this property made it necessary to examine the possible history of the sequence, ie the calculation of the sequence "backwards". This immediately led to the discovery of sequences of exciting new features and relationships, but again proved to be a generalization. Here the rule changed like this: F*(n) = F*(n-2) - F*(n-1).

Because subtraction is as much a hassle operation as addition, we immediately get an infinite sequence (meaning all integers) in both directions, which is the original Fibonacci sequences in one direction and the same in the other direction.

It is also worth noting that this "mixing" is already due to the choice of the first two members of the sequences (or at least one negative number (which is not prevented by the impetus of generalizations).

Whichever direction we look at these sequences now, there are some very interesting rules at the markings of the sequences: each sequences has a peculiar "breakpoint, and from then on the sequences is sharply increasing or sharply decreasing in one direction, and the signs in the other direction alternate regularly.

It was this sign change that led me to "tile a sign change into the basic formula itself, so I wrote this formula:

F^S(n)=((-1)^(F^S(n-2)+F^S(n-1)))* F^S(n-2)+F^S(n-1)

To my surprise, this sequence turned out to be periodic, and not only with the original 1, 1 starting members, but with other starting numbers in every subsequent attempt. This, in the case of a standard mathematical formula (using only basic operations as addition and subtraction, multiplication and division, power and root subtraction), to say the least, was surprising.

Here are some examples:

Interestingly, the last column shows that the sequence of numbers immediately becomes negative and thus, underwater, floats to the other side of the new cycle.

Yes, but some new experiments have brought to light the lability of this new feature, or rather the Achilles heel. What happens is that if both of the novice members - and only then - lose the periodic nature of the sequences, they will switch back to the "regular" Fibonacci sequences. However, it should be noted that it takes the multiplication with any number very easily, since ac + bc = (a + b)c. Therefore, every Fibonacci sequences is considered a "basic" sequences multiplied by the largest common divisor of a and b. So if the "basic", or "serious" sequence, the two starting members can't be even at the same time (since 2 is still part of the largest common divisor). So, in fact, such an unfortunate clone would be able to fall out of the periodic character. Anyway, it's good to know about this Achilles heel. Otherwise, it is not difficult to understand why this "setback" occurs in the classic form: it is simply that for a pair of even pairs, the sum of each of the two adjacent members of the sequences is even, which generically blocks (unifies) the (-1) prefix in the training rules of the sequences.

But even with this weird exception, the basic case is surprising and interesting. A number of things are easy and quick to determine, but there are likely to be other exciting questions to ask.

One of the most important statements is that the length of the period is constant and it is 6 members. It does not depend on the sign of the initial members, the size of the members, whether b is greater or lesser, or even equal to a.

It is also a constant property that the cycle changes signs at least twice, i.e. at least one member has a different sign than the other. However, as shown in the examples above, the sign change can span up to four members.

Leonardo Pisano (Fibonacci), 1175-1250

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Great calculator family

Recently, I wanted to do a certain calculation with extremely large numbers. I searched for a suitable tool on the net and found a great collection of calculators to my great satisfaction.

Here is how they present themselves:

About Us

We are a group of IT professionals enthusiastic in creating quality free tools and content on the internet. The main purpose of this website is to provide a comprehensive collection of free online calculators for the ease of public use. This site was launched on calculators.info first in 2007. In 2008, we migrated to calculator.net.

The calculators on this site were grouped into 4 sections: financial, fitness & health, math, and others. All of the calculators were developed in-house. Some calculators use open-source JavaScript components under different open-source licenses. More than 90% of the calculators are based on well-known formulas or equations from textbooks, such as the mortgage calculator, BMI calculator, etc. If formulas are controversial, we provide the results of all popular formulas, as can be seen in the Ideal Weight Calculator. Calculators such as the love calculator that are solely meant for amusement are based on internal formulas. The results of the financial calculators were reviewed by our financial advisors, who work for major personal financial advising firms. The results of the health calculators were reviewed and approved by local medical doctors. More than 95% of the descriptive content was developed in-house with a small amount of content taken from wikipedia.org under the GNU Free Documentation License. The descriptive content of the financial calculators was created and reviewed by our financial team. The descriptive content of the health calculators was reviewed by local medical doctors.

Calculator.net

Great project, great work!

Thanks for.

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Pi as Music

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Lorentz-friend numbers

n and m natural numbers Lorentz-friends if the Lorentz-transformation of the two Egyptian fractions 1/n and 1/m is also an Egyptian fraction.

As a reminder: Lorentz-transformation of n and m is (n+m)/(+nm).

So n and m are Lorentz-friend if and only if exists natural number k that

(1/n+1/m)/(1+1/nm) = 1/k

Like the light, 1 is Lorentz-friend with all naturals (the Lorentz transformation of 1 and m is 1)! There is concern about this kind of indiscriminate friendship, so we would rather call it pseudo-friendship and ignore it simply when talking about Lorentz friendship.

A perfect antipode is 2 that has no true Lorentz friends.

Another extremely interesting phenomenon is that every odd number and the following odd are Lorentz friends. In fact:

(1/2n-1)+1/(2n+1))/(1+1/(2n-1)(2n+1)) = 1/n

It follows from the above that every odd number greater than 3 has at least two Lorentz friends.

Is it possible to know how many Lorentz friends a natural number has? What relationships and characteristics can be established?

Every natural number greater than 2 seems to have at least one, every odd number to has at least two Lorentz friends, but not always more. However, there are many examples where a number has 3 or more Lorentz friends.

For example 16 have 3 friends: 35, 69 and 239. The nearby 18 only have one friend: 305.

There are bigger achievements. So 31 have 13 friends: 9, 17, 29, 33, 49, 65, 89, 129, 161, 209, 289, 449, 929.

We can define a sequence a(n) that gives n the smallest Lorentz friend which largest that n. Clearly, if n is odd a(n)=n+2, it can make the sequence a bit dull. Let's hope that some interesting things will come. For the sake of order, it should be agreed that a(1)=1, and if n has no friends, then a(n)=0. So the sequence looks like this:

1, 0, 5, 11, 7, 29, 9, 13, 11, 23, 13, 131, 15, 25, 17, 35, 19, 305, 21, 37, 23,…

To be continued.

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