Primax - International Math Competition

 The editors of Acta Mundi and the Changing World announce an international mathematics competition

PRIMAX

according to the following conditions.

1. The purpose of the competition is to find functions that give prime numbers in a maximum number. In the use of the term in this competition, a function is a closed arithmetic formula with one variable interpreted on the natural numbers.

2. Anyone may enter the competition by sending an e-mail to primax@valtozovilag.hu with the formula of the function and any other information necessary to participate until the end of November of the current year. Participation in the competition requires a net payment of 1001 HUF to the editorial bank account.

3. It is irrelevant whether the subject formula is original or already known in mathematics. At the same time, it will be appreciated if the participant indicates the origin of the existing formula.

4. The winner of the competition is the one who submits the formula that produces the highest number of prime numbers. The evaluation is done in two categories.

A. Total result: the number of prime numbers in the first 100 values. In case of equality, we take into account the number of prime numbers between the first 200 values, etc.

B. Initial result: counts the number of consecutive prime numbers starting from the first value.

5. The winners of the competition will receive a diploma and a cash reward equal to 80% of the participation fees and sponsors' donations. If in one year there is no better result than the previous year, the new contributions will increase the reward for next year.

6. Winners of both categories will be announced separately, their rewards are equal. If several participants submit the same formula, the first one who submitted the formula correctly will be considered the winner.

7. The date of the announcement of the results is December 13 of the current year, the first opportunity on December 13, 2019.

8. Applicants must agree that the resulting sequence of the winning formula will be added to the OEIS, if it is not part of it, otherwise the editors will be allowed to submit it.

9. Only international and Hungarian laws will be applicable to the competition.

Budapest, September 20, 2019

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One interesting sequence

 

1, 2, 1, 3, 2, 4, 5, 3, 6, 7, 4, 10, 8, 12, 12, 13, 4, 16, 17, 3, 18, 19, 4, 22, 14, 19, 8, 26, 16, 28, 24, 29, 4, 32, 33, 25, 32, 36, 36, 37, 4, 40, 38, 25, 40, 41, 6, 45, 44, 39, 38, 33, 42, 47, 8, 54, 54, 55, 48, 57, 42, 55, 52, 51, 48, 63, 60, 63, 62, 39…

The sequence’s formula is a(1)=1, a(2)=2, for n>2 a(n)=n-largest prime which divides a(n-1). For “the largest prime which divides n we can use the LPD(n) notation.

For example, let n=6. a(5)=2. Largest prime, which divides 2, is 2. So a(6)=6-2=4.

Is this sequence interesting?

Everyone has the right to his own opinion, and therefore no one is responsible.

I thing that this sequence is very interesting because a(n)=n-LPD(a(n-1)), and no a(n)=n-LPD(n-1). In the latter case, we obtain a similar but less exciting sequence. Because of this peculiarity, I would call the original sequence a “kalach” sequence.

One first observation: if n is a prime then a(n) is often even, but there is at least one exception: a(19) = 17.  Conversely, it appears that if a(n) is a prime, then n is even.

According to my intention, I will report on other interesting things here, so the note may be expanded or modified in the future.

If you have any comments or observations, feel free to let us know.

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Question – why interesting?

Why interesting

387 420 489

Answer

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The numbers

 

My new book under the title (ambitious?) The numbers this Tuesday, February 5, 2019 published.

I am very happy to have repaid some - very little - of my debt to my parents who had made possible my university studies in Budapest, my first teachers of mathematics in primary and French high school in Varna, all the same towards my teachers from the university, among which there are a dozen first-class mathematicians, and finally to Mathematics itself, love, I would say - forever and forever, now I await the judgment of the readers.

I dare to propose this small volume (of 128 pages) to the attention of the laity and all the same to that of professionals. I hope that "all will be served according to their taste".

Here is the table of contents:

I. The numbers of nature

1. "Borko, bring me apples!"

2. Numbers, operations, structures

3. Numbers and other attractions

II. The numbers of the ratio

1. Chipping the numbers

2. Negative Numbers - Victory of Positive Thinking

3. The peak of the ratio: the names of the numbers and the systems

III. The numbers of transcendence

1. Ratio + irratio = Reality?

2. Irratio, at the complex

3. The ambassadors of transcendence

IV. The numbers of the mystic

1. Gnosis and mystery

2. The holy numerology

3. Paradoxes, sophistry, Great secret

V. Numbers of mathematics

1. The strange world of mathematics

2. The strange history of mathematics

3. The number of dethroned

VI. The numbers of the future

1. Go, number!

2. Models and dynamics

3. The language of numbers

VII. Number files

And when could you read this little book in English to know all that matters about numbers?

Hope that soon. I am looking for a publisher.

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A mathematical hypothesis

According to my hypothesis, any natural number, so any positive integer can be produced as the sum of the square of an integer and a prime or at most two prime.

In this case, it is important (otherwise the conjecture is not true) to include 1 as prime number. (My personal point of view is that 1 should be considered as prime, but keeping the necessary distinctions as appropriate: primes bigger than 1 or bigger than 2. Out of respect for the tradition, it is reasonable to call prime numbers the primes bigger than 1, big primes the primes bigger than 2, in the end absolute prime numbers the set of all primes and the 1. 1 merit the respect, not only because that it is the ancestor of all numbers, but because it fulfills perfectly the most natural definition of a prime number: p is prime number, if from a × b = p follows that a = 1 and b = p , or the opposite.)

Let us see some examples that obviously only confirm but do not prove the conjecture.

1 = 1 + 0²

2 = 1 + 1²

3 = 3 + 0²

4 = 3 + 1²

5 = 5 + 0²

6 = 5 + 1²

7 = 7 + 0²

8 = 7 + 1²

9 = 5 + 2²

10 = 1 + 3²

100 = 19 + 9²

1000 = 919 + 9²

10000 = 199 + 99²

123 = 2 + 11²

1234 = 1009 + 15²

12345 = 1109 + 106²

It is easy to spot that in some cases the number can be produced in different ways. For example:

12345 = 1109 + 106² = 2741 + 98²

Interestingly, most natural numbers can be produced as the sum of a square and one prime number, but there are “hard” cases that can only be solved with two prime numbers.

So far, according to calculations – partly reported by others – from 1 to 100 there are six exceptions.

25 = 5 + 19 + 1²

34 = 11+ 23 + 0²

58 = 3 + 53 + 1²

64 = 11 + 53 + 0²

85 = 31 + 53 + 1²

91 = 2 + 89  + 0²

The "difficult" cases between 100 and 1000: 121, 130, 169, 196, 214, 289, 324, 370, 400, 526, 529, 625, 676, 706, 771, 784, 841. It can be seen that their Frequency decreases fastly, but it is unlikely they will "run out". I suppose for example that the numbers of the form

2^2(2k+1)

are all difficult cases. (It would certainly be fitting to give a name to both types of natural numbers.)Was already made this hypothesis, I do not know. Among other things, that is why I did not want to give him a name (which is not important, but practical to mention it).

Moreover, I am convinced that this statement – unfortunately – cannot be proved in the same way as the famous Goldbach's conjecture (which, I think, has nothing to do with Gödel's "incompleteness" theorem, but is due to the mystery of the prime numbers).

This is our world.

March 3, 2019, Todor Simeonov

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The Simi numeral system

It is superfluous to present the numerical system practically the only one used today in the world. It is also well-known that computing uses a same – binary – system and that any other system can be set up. If we want to evaluate the various numeral systems, we could propose some – somewhat subjective – requirements:

- be easy to understand and learn,

- be comfortable for adding numbers,

- be comfortable for multiplying numbers,

- be able to quickly identify certain properties of the number.

We might ask ourselves, how much this makes sense, since we do not really know a new alternative numeral system. However, in 2019 the situation changed: the Simi numeral system (SNS) was born!

The rule is that we write down the canonical form of the number (in descending order of the primes), but with all the smaller prime numbers (with a zero power), and then we sort the powers one after the other. This sequence is the number in the Simi numeral system. Eg. 20 = 5¹×3⁰×2². Thus, in the Simi system 20 is described: 102. The Fundamental theorem of arithmetic guarantees that in this form all natural numbers can be written by one and only one way.

Decimal	Simi	Decimal	Simi
1	0	11	10 000
2	1	12	12
3	10	13	100 000
4	2	14	1001
5	100	15	110
6	11	16	4
7	1000	17	1 000 000
8	3	18	21
9	20	19	10 000 000
10	101	20	102

What's the point, what would be the advantage of this numeral system? Cannot overlook some extraordinary, very useful features of the new system. A huge, unmatched advantage: multiplying the numbers written in this system (which, let's face it, especially if the numbers are very large, is a macerated task), can be done here by simple addition. Remaining in the table above: multiples of 4 and 5 are 20, in the Simi system the same is 2 plus 100, which is 102. One can imagine how pleased computers would be to multiply millions of times per second for more serious tasks.

There is a natural consequence of this advantage: raising a number to a square or any power becomes a simple multiplication. Decimal: 4 on the second 16, on the Simi system 2 on the second 4. True, adding here is a harder matter. Continuing with the previous example, the sum of 2 and 100 is 20! Why, who knows?

But the Simi system has another incredible advantage. Since primes have appeared, mathematicians have been overly depressed by their ignorance of the regularity of the appearance of primes, and we have no formula that gives us the 100th prime, if desired. It is now resolved in this form forever and for all primes. The hundredth prime is simply 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

Here is the moment when skeptics can speak up: what do we know from the above inscription on the hundredth prime? Why, we could answer that, and what would we know if the same thing were written in decimal? (The debate is by no means uninteresting, but this is not the moment).

Let's see another interesting chart.

Decimal	Simi	Decimal	Simi
2	1	1	0
4	2	10	101
8	3	100	202
16	4	1 000	303
32	5	10 000	404
64	6	100 000	505
128	7	1 000 000 000	606
256	8	10 000 000 000	707
512	9	100 000 000 000	808
1024	(10)	1 000 000 000 000	909

The calculation of the smallest common multiple and the largest common divisor is also very simple: an operation similar to traditional addition. We write the two numbers one under the other, and and choose for the result in the first case the largest, in the second the smallest digit. Incidentally, in this system, divisibility by a number is no longer an issue.

The Simi system seems to have a serious weakness. How should one describe a number that has a prime with greater power the 9? Undoubtedly, this interferes with the incredible convenience of the numerical system, but the problem is not insolvable. A possible solution would be to put a power greater than 9 in parentheses. According to this, the infamous Googol number, 10¹⁰⁰, would be described in Simi as (100)0(100).

Otherwise, this technique can also simplify the writing of the hundredth prime. It would look like this: 1((99) 0).

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