When the number of divisors is greater than the number itself

 

The fact that a prime can be written as a nontrivial product of two complex integers is certainly very surprising to many, it may even be shocking.

Surprising or not, the fact is true and it would be right to go around. Perhaps not all primes “betray” the ancient classic character of not divisible building blocks. It can be seen quickly. that every prime is a “traitor”. What's more. We see that every prime number can be written in surprisingly many ways as a nontrivial product. (By the way, it would be absolutely right for someone to despise the mention of surprise, since it really isn’t a mathematical concept).

Let's see an example! The number of nontrivial divisors of the popular 13 is… 14!

 

-13+0i, -3-2i, -3+2i, -2-3i, -2+3i, -1+0i, 0-13i, 0-1i, 0+1i, 0+13i, 2-3i, 2+3i, 3-2i, 3+2i

 

But if 13 is capable of this, why be surprised that 12 has 38 nontrivial divisors among complex integers. Now, ready for anything, let’s look at the number of nontrivial divisors of all natural numbers between 1 and 100! These:

 

3, 10, 6, 18, 14, 22, 6, 26, 10, 46, 6, 38, 14, 22, 30, 34, 14, 34, 6, 78, 14, 22, 6, 54, 34, 46, 14, 38, 14, 94, 6, 42, 14, 46, 30, 58, 14, 22, 30, 110, 14, 46, 6, 38, 46, 22, 6, 70, 10, 106, 30, 78, 14, 46, 30, 54, 14, 46, 6, 158, 14, 22, 22, 50, 62, 46, 6, 78, 14, 94, 6, 82, 14, 46, 70, 38, 14, 94, 6, 142, 18, 46, 6, 78, 62, 22, 30, 54, 14, 142, 30, 38, 14, 22, 30, 86, 14, 34, 22, 178

 

I definitely have a feeling that we still have a lot of interesting things to do here.

 

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A dramatic encounter

 

Each prime number can be written as a non-trivial product of two integers, that is, so that neither factor is 1 - I had to notice that.

I admit it was a special experience. At first hearing, this is truly an apocalypse in classical number theory.

The shock is resolved by a small detail of the situation: the two integers whose product gives the prime are complex integers:

(4+5i)(4-5i) = 41

I found many primes among similar, i.e. (k + 5i) (k-5i) numbers, their number being infinite (but this should remain a hypothesis for the time being). These:

 

29, 41, 61, 89, 281, 349, 509, 601, 701, 809, 1049, 1181, 1321, 1789, 2141, 2729, 3389, 4649, 5209, 5501, 5801, 8861, 9241, 9629, 10429, 11261, 11689, 12569, 15401, 15901, 17449, 17981, 18521, 19069, 21341, 21929, 23741, 24989, 26921, 27581, 33149, 39229, 40829, 41641, 42461, 45821, 46681, 52009, 53849, 55721, 59561, 68669, 71849, 79549, 80681, 86461, 87641, 91229, 94889, 97369, 98621, 99881, 101149, 107609, 111581, 112921, 114269, 116989, 118361, 126761, 128189, 133981, 135449, 139901, 145949, 147481, 149021, 153689, 156841, 158429, 169769, 173081, 174749, 179801, 181501, 186649, 190121, 195389, 197161, 198941, 204329, 209789, 226601, 228509, 234281, 252029, 254041, 264221, 266281…

 

Similar sequences can be found if any other number is entered in place of 5 in the above formula. Interestingly, in the first four cases we all get a series that is known in OEIS - but not because of the property indicated here, but by referring to the expanded form of the related formula:

5, 13, 17, 29, 37, 41, 53, 61 - https://oeis.org/A002144

2, 5, 17, 37, 101, 197, 257 - https://oeis.org/A002496

 5, 13, 29, 53, 173, 229 - https://oeis.org/A005473

13, 73, 109, 409, 1033 - https://oeis.org/A138353

17, 41, 97, 137, 241, 457 - https://oeis.org/A243451

 

After the above observation, a number of exciting questions arose in me. I have already found the answer to most of these. I will share these in a subsequent post.

 

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The legend of the prime numbers

 

There was once a legend that 2, 3, 5, 7, 11 and many other numbers are prime, that is, they cannot be divisible by an integer other than 1 and itself.

The legend still holds true today, but I have news: it’s not true.

Unfortunately, now I can't say for sure whether all legendary primes are complex numbers, or or whether there may be new real primes not yet discovered.

I’ll think, and if I have news, I’ll report.

But in the meantime, I’ll share details about the legend unveiling soon.

 

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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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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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