This article is "2022 The fourth Mathematical Culture Essay Contest Activities
Note: Supplementary of the attached table 1
An accidental discovery - complete number and its gene construct sequence
Author: Liu Jian
Work number: 091
Complete number is also called perfect number, which is said to be discovered by young scholars from the Pythagoras School of Ancient Greek . This school's view believes that numbers are the root of all things, and the laws of mathematics are the laws of the universe. They noticed that number 6 has a characteristic, which is equal to the sum of factors other than itself, that is, 6=1+2+3. Call this type of number a perfect number. " Geometry Original " first recorded 6, 28, 496, 8128. These four perfect numbers and proved. If it is a prime number, it is a complete number.
Until the Western Renaissance, after more than 2,000 years of historical progress, people discovered three complete numbers one after another, namely 33550336 and 858986 9056, 137438691328.
In the late 18th century, the famous mathematician Euler found the 8th complete number, 2305843008139952128, and further proved that if it is a prime number, it is a unified simplified expression of all known complete numbers, which establishes a close connection between the complete number and the Mason prime number. In order to find new large prime numbers and complete numbers, Breaking new paths. Over the past 200 years, with the continuous development of computer technology, people have accelerated the pace of finding complete numbers. So far, after more than 2,500 years, people have found a total of 51 Mason prime numbers, that is, people have discovered 51 complete numbers so far.
complete numbers are very strange numbers. In addition to being equal to themselves, they also have some special properties.
(1) Complete numbers are triangle numbers, which can be expressed as the sum of continuous natural numbers.
(2) They are all cubic sums of continuous odd numbers, except for 6, and they can also be expressed as the sum of continuous positive integer powers of 2.
(3) Except for 6, their root of numbers - that is, the numbers on each bit are added repeatedly until they become a single number, and this single number must be 1.
(4) Except for factor 1, the sum of the reciprocals of all factors of each complete number is equal to 1.
(5) Complete numbers end with 6 or 8. If they end with 8, it must end with 28.
(6) Their binary expressions have similar structural forms.
110
11100
111110000
111111100000
…
Another thing worth noting is that all known complete numbers are even complete numbers, and whether there are odd complete numbers is still an unsolved mystery. The above is the basic knowledge about complete that you can know after checking relevant information.
18th century mathematician Euler uses his proof of safe numbers to tell us. Complete numbers and Mason numbers have a close connection, and finding new Mason prime numbers becomes the prerequisite for discovering new complete numbers. But finding new large prime numbers is a big problem to be solved. Therefore, the pace of finding new complete numbers is difficult and slow, but in recent years, with the help of the power of rapidly developing computers, some progress has been made.
In June 2001, when I was sorting out old publications, I saw an intellectual question about "Find Complete Numbers" from the second issue of "Mystery" in 1998. What is a complete number? How to find the complete number? As a third-year old, his math grades were still good at that time, but looking back at what he learned over the years, there was nothing to do with it. Although he is already a person who knows the destiny, what is the perfect number? How to find complete numbers without knowing anything. After finding the answer in the publication, it checks according to the condition that the complete number is equal to the sum of factors other than itself. As a result, the complete number given by the answer is 33550326, and the decomposition factor appears in the decomposition factor, which is obviously wrong.
By factoring the arrangement and combination of on the known 4 complete numbers, I quickly found out that the fifth complete number is 33550336, and found the structural rules in it to construct a new sequence with the common properties of many complete numbers. When people saw the relevant records and found the fifth complete number, which lasted more than 2,000 years, I felt the significance and value of this discovery. After repeated learning and continuous exploration, it has lasted for more than 20 years and has achieved some gratifying results.
Reviewing this matter at first, it was due to curiosity. Then it’s like planting beans and getting melons, and it’s unexpectedly harvested. The original intention is to find a complete number, but what is found is a new sequence with the complete number 6 as the first term and has some special properties.
6, 28, 496, 8128 are the earliest four complete numbers recorded in "The Origin of Geometry" and are also the first four terms of this sequence. They have some of the same special properties and they all have a similar factor decomposition arrangement structure. In addition, their factor decomposition arrangement forms the equal ratio sequence with two terms equal and the ratio is 2. (Appendix 1) I call this: symmetrical overlap structure.
Anyone who has studied chemistry knows that the structure of a substance determines its properties. Similarly, the structure of a sequence can also determine its properties. The first complete number 6 and its factor decomposition 1, 2, 3, 6 is a standard symmetrical overlap structure. 1, 2 and 3, 6 are two equal-scale sequences with equal terms and ratio 2. This special property determined by the structure, maintains, passes and extends between numbers with the same structure, forming a new sequence of numbers.
first digit complete number 6, and its factors 1, 2, 3, 6, equals the sum of factors outside oneself 1+2+3=6, which is concise and beautiful, reminding people of the "Tao Te Ching" by Laozi : Tao gives birth to one, one gives birth to two, two gives birth to three, and three gives birth to all things. ...Is everything equal to 6? ...
Look at the symmetrical overlapping structure composed of 1, 2, 3, 6, four numbers and two sequences, and you can also associate the genetic material composed of four bases and two spiral curves. ...They will convey their inner properties, and they are all like this four and two structures. ...Is it just a coincidence?
is a bit far from it, so let's get back to the topic.
6, 28, 496, 8128, 130816, 2906128, 33550336, 536854528,... These numbers contain both complete numbers and non-complete numbers, but because they have many common special properties and have a factorization symmetric overlap structure, they form a new sequence - a complete number gene constructed sequence. (Appendix 2) The simplest expression of the sequence
: When n is a natural sequence, the sequence corresponds to the natural sequence one by one, indicating that it has infiniteness. In this way, all known even complete numbers will be found here, and new complete numbers will inevitably appear on this sequence in the future.
Complete number is like the pearls scattered in the math forest, it is very rare and rare. Euler's formula has proved that Mason's prime number and complete number correspond one by one. On the complete number gene construct sequence, the factor decomposition mostly forms Mason numbers, not Mason prime numbers. Neither prime number. It is formed correspondingly. It's definitely not a complete number. But these non-complete numbers with the same construction are like a long line that connects all (pearls) complete numbers together to form a perfect sequence.
constructs sequences based on complete numbers. We can supplement and improve the existing knowledge related to complete numbers.
(1) Complete number and its gene construct sequence are all ending with 6 or 8. If it ends with 8, it must end with 28. If it ends with 6, there will be 16, 36, 56, 76, 96, and partition cycles.
(2) Except for the first complete number 6, all numbers on this sequence are integer multiples of 4, and all are divided into 6 and more than 4.
(3) We call such a number the Mason number. If it is a prime number, it is called a Mason prime number. In this sequence, a series of Mason numbers formed by factorization arrangement will have a 7, 4, and 1 cycle phenomenon. If it is a Mason prime number, it must be a 6n+1 prime number.
complete number gene construct sequence is the simplest expression, which looks different from the Euler complete number formula. In essence, it is compatible and consistent. Through the (Appendix Table 1) model, we can intuitively see that they all contain symmetrical overlap structures. The difference is that one has the premise, which is the Mason prime number, corresponding to a complete number. A without premise, corresponding to a natural sequence, forms an infinite sequence.
For Euler's complete number formula (except for the complete number corresponding to the Mason prime number) If n is a natural sequence, the overall result will be disordered.If n is an odd number sequence headed by 3, it is surprisingly surprising, and the result is exactly the same as the complete number gene construct sequence. (Appendix 3)
natural sequence and odd number sequence, each correspond one by one to the complete number gene construct sequence. This not only proves its infiniteness, but also shows the inevitability and rationality of its existence.
is unprecedented infinity sequences starting with 4 complete numbers, containing symmetrical overlapping structures, and having multiple common and special properties. Through the attached diagrams and models, we can see the order and perfection of the digital world. There is symmetry and balance here, there are gradual changes and mutations. There are numbers like cell division , and there are also phenomena similar to genetic inheritance.
Genetic inheritance is a unique phenomenon in the life world. Does the highly abstract digital world also have genetic phenomena? …
In the process of inorganic chemistry developing towards organic chemistry, people have realized the influence of the structure of molecules on chemical properties. "The molecular structural formula not only represents the number of atoms, but also represents the arrangement order and binding relationship of various atoms in the molecule. The structure of their composition is closely connected with their chemical properties."
The same is true, what you see here is: the arrangement order and bonding relationship of various factors in the sequence are closely linked to the special properties of the complete number they maintain. Here the role of structure on properties is decisive.
Is it practical? Of course, there are, detecting the computing power of a computer and multiple verifications, which can be directly used.
Another intuition: there are many ways to compile various passwords. …
Finally: I would like to express my gratitude to the three friends of Zhang Difei who gave inspiration and correction, Zhu Changhe and Ji Guilin who gave support and help. This article is "2022 The fourth Mathematical Culture Essay Contest Activities Note: Supplementary of the attached table 1 An accidental discovery - complete number and its gene construct sequence Author: Liu Jian Work number: 091 Complete number is also called perfect number, which is said to be discovered by young scholars from the Pythagoras School of Ancient Greek . This school's view believes that numbers are the root of all things, and the laws of mathematics are the laws of the universe. They noticed that number 6 has a characteristic, which is equal to the sum of factors other than itself, that is, 6=1+2+3. Call this type of number a perfect number. " Geometry Original " first recorded 6, 28, 496, 8128. These four perfect numbers and proved. If it is a prime number, it is a complete number. Until the Western Renaissance, after more than 2,000 years of historical progress, people discovered three complete numbers one after another, namely 33550336 and 858986 9056, 137438691328. In the late 18th century, the famous mathematician Euler found the 8th complete number, 2305843008139952128, and further proved that if it is a prime number, it is a unified simplified expression of all known complete numbers, which establishes a close connection between the complete number and the Mason prime number. In order to find new large prime numbers and complete numbers, Breaking new paths. Over the past 200 years, with the continuous development of computer technology, people have accelerated the pace of finding complete numbers. So far, after more than 2,500 years, people have found a total of 51 Mason prime numbers, that is, people have discovered 51 complete numbers so far. complete numbers are very strange numbers. In addition to being equal to themselves, they also have some special properties. (1) Complete numbers are triangle numbers, which can be expressed as the sum of continuous natural numbers. (2) They are all cubic sums of continuous odd numbers, except for 6, and they can also be expressed as the sum of continuous positive integer powers of 2. (3) Except for 6, their root of numbers - that is, the numbers on each bit are added repeatedly until they become a single number, and this single number must be 1. (4) Except for factor 1, the sum of the reciprocals of all factors of each complete number is equal to 1. (5) Complete numbers end with 6 or 8. If they end with 8, it must end with 28. (6) Their binary expressions have similar structural forms. 110 11100 111110000 111111100000 … Another thing worth noting is that all known complete numbers are even complete numbers, and whether there are odd complete numbers is still an unsolved mystery. The above is the basic knowledge about complete that you can know after checking relevant information. 18th century mathematician Euler uses his proof of safe numbers to tell us. Complete numbers and Mason numbers have a close connection, and finding new Mason prime numbers becomes the prerequisite for discovering new complete numbers. But finding new large prime numbers is a big problem to be solved. Therefore, the pace of finding new complete numbers is difficult and slow, but in recent years, with the help of the power of rapidly developing computers, some progress has been made. In June 2001, when I was sorting out old publications, I saw an intellectual question about "Find Complete Numbers" from the second issue of "Mystery" in 1998. What is a complete number? How to find the complete number? As a third-year old, his math grades were still good at that time, but looking back at what he learned over the years, there was nothing to do with it. Although he is already a person who knows the destiny, what is the perfect number? How to find complete numbers without knowing anything. After finding the answer in the publication, it checks according to the condition that the complete number is equal to the sum of factors other than itself. As a result, the complete number given by the answer is 33550326, and the decomposition factor appears in the decomposition factor, which is obviously wrong. By factoring the arrangement and combination of on the known 4 complete numbers, I quickly found out that the fifth complete number is 33550336, and found the structural rules in it to construct a new sequence with the common properties of many complete numbers. When people saw the relevant records and found the fifth complete number, which lasted more than 2,000 years, I felt the significance and value of this discovery. After repeated learning and continuous exploration, it has lasted for more than 20 years and has achieved some gratifying results. Reviewing this matter at first, it was due to curiosity. Then it’s like planting beans and getting melons, and it’s unexpectedly harvested. The original intention is to find a complete number, but what is found is a new sequence with the complete number 6 as the first term and has some special properties. 6, 28, 496, 8128 are the earliest four complete numbers recorded in "The Origin of Geometry" and are also the first four terms of this sequence. They have some of the same special properties and they all have a similar factor decomposition arrangement structure. In addition, their factor decomposition arrangement forms the equal ratio sequence with two terms equal and the ratio is 2. (Appendix 1) I call this: symmetrical overlap structure. Anyone who has studied chemistry knows that the structure of a substance determines its properties. Similarly, the structure of a sequence can also determine its properties. The first complete number 6 and its factor decomposition 1, 2, 3, 6 is a standard symmetrical overlap structure. 1, 2 and 3, 6 are two equal-scale sequences with equal terms and ratio 2. This special property determined by the structure, maintains, passes and extends between numbers with the same structure, forming a new sequence of numbers. first digit complete number 6, and its factors 1, 2, 3, 6, equals the sum of factors outside oneself 1+2+3=6, which is concise and beautiful, reminding people of the "Tao Te Ching" by Laozi : Tao gives birth to one, one gives birth to two, two gives birth to three, and three gives birth to all things. ...Is everything equal to 6? ... Look at the symmetrical overlapping structure composed of 1, 2, 3, 6, four numbers and two sequences, and you can also associate the genetic material composed of four bases and two spiral curves. ...They will convey their inner properties, and they are all like this four and two structures. ...Is it just a coincidence? is a bit far from it, so let's get back to the topic. 6, 28, 496, 8128, 130816, 2906128, 33550336, 536854528,... These numbers contain both complete numbers and non-complete numbers, but because they have many common special properties and have a factorization symmetric overlap structure, they form a new sequence - a complete number gene constructed sequence. (Appendix 2) The simplest expression of the sequence : When n is a natural sequence, the sequence corresponds to the natural sequence one by one, indicating that it has infiniteness. In this way, all known even complete numbers will be found here, and new complete numbers will inevitably appear on this sequence in the future. Complete number is like the pearls scattered in the math forest, it is very rare and rare. Euler's formula has proved that Mason's prime number and complete number correspond one by one. On the complete number gene construct sequence, the factor decomposition mostly forms Mason numbers, not Mason prime numbers. Neither prime number. It is formed correspondingly. It's definitely not a complete number. But these non-complete numbers with the same construction are like a long line that connects all (pearls) complete numbers together to form a perfect sequence. constructs sequences based on complete numbers. We can supplement and improve the existing knowledge related to complete numbers. (1) Complete number and its gene construct sequence are all ending with 6 or 8. If it ends with 8, it must end with 28. If it ends with 6, there will be 16, 36, 56, 76, 96, and partition cycles. (2) Except for the first complete number 6, all numbers on this sequence are integer multiples of 4, and all are divided into 6 and more than 4. (3) We call such a number the Mason number. If it is a prime number, it is called a Mason prime number. In this sequence, a series of Mason numbers formed by factorization arrangement will have a 7, 4, and 1 cycle phenomenon. If it is a Mason prime number, it must be a 6n+1 prime number. complete number gene construct sequence is the simplest expression, which looks different from the Euler complete number formula. In essence, it is compatible and consistent. Through the (Appendix Table 1) model, we can intuitively see that they all contain symmetrical overlap structures. The difference is that one has the premise, which is the Mason prime number, corresponding to a complete number. A without premise, corresponding to a natural sequence, forms an infinite sequence. For Euler's complete number formula (except for the complete number corresponding to the Mason prime number) If n is a natural sequence, the overall result will be disordered.If n is an odd number sequence headed by 3, it is surprisingly surprising, and the result is exactly the same as the complete number gene construct sequence. (Appendix 3) natural sequence and odd number sequence, each correspond one by one to the complete number gene construct sequence. This not only proves its infiniteness, but also shows the inevitability and rationality of its existence. is unprecedented infinity sequences starting with 4 complete numbers, containing symmetrical overlapping structures, and having multiple common and special properties. Through the attached diagrams and models, we can see the order and perfection of the digital world. There is symmetry and balance here, there are gradual changes and mutations. There are numbers like cell division , and there are also phenomena similar to genetic inheritance. Genetic inheritance is a unique phenomenon in the life world. Does the highly abstract digital world also have genetic phenomena? … In the process of inorganic chemistry developing towards organic chemistry, people have realized the influence of the structure of molecules on chemical properties. "The molecular structural formula not only represents the number of atoms, but also represents the arrangement order and binding relationship of various atoms in the molecule. The structure of their composition is closely connected with their chemical properties." The same is true, what you see here is: the arrangement order and bonding relationship of various factors in the sequence are closely linked to the special properties of the complete number they maintain. Here the role of structure on properties is decisive. Is it practical? Of course, there are, detecting the computing power of a computer and multiple verifications, which can be directly used. Another intuition: there are many ways to compile various passwords. … Finally: I would like to express my gratitude to the three friends of Zhang Difei who gave inspiration and correction, Zhu Changhe and Ji Guilin who gave support and help. 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