Question – This number wit fewer difitd?

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Alone with Numbers

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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

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We recommend it to everyone's attention.

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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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Answer – Why interesting?

The 387 420 489 2 is the largest number that can be expressed by two digits:

387 420 489 = 9⁹

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In the Mood for Love - For quiet thinking

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