The perfect digital mystery: one of the oldest fields in mathematics, many problems have not been solved until now.

The full number is an integer equal to the sum of its factors, and these "perfect" numbers fascinate mathematicians, but finding a full number is difficult to find.
search for full numbers began in ancient Greece, but so far we have found only 51 such figures.
a long time ago, mathematicians began to study the nature of integers and the relationshipbetween integers.
study of these numbers has also evolved into a major branch of mathematics, the theory of numbers, one of the oldest fields in mathematics.
it has a long history and contains many profound and fascinating problems, many of which have not been solved until now.
in one of these questions, one is related to the full number.
the exact number is an integer equal to the sum of its factors, such as 6 with a factor of 1, 2, 3, and 1 plus 2 plus 3, and for example, 28 can be divisible by 1, 2, 4, 7, 14, and 1 plus 2 plus 4 plus 14 x 28.
numbers like 6 and 28 are called full numbers (not perfect numbers, complete numbers).
these "perfect" numbers fascinate mathematicians because, in their easy-to-understand definitions, they are complex and mysterious - it's not hard to define such a full number, but it's hard to find a full number.
search for full numbers began in ancient Greece, but so far we have found only 51 such figures.
the biggest one was discovered in 2018, and I can't show here how much that number is because it has nearly 50 million digits.
the story of O'Reed's full number began more than two thousand and three hundred years ago and appeared in Euclid's mathematical book, Geometric Originals.
in the last proposition of Chapter IX of Geometry, Euclidean gives for the first time a way to find a full number: "If any set of numbers starts from the smallest integer and expands continuously in two times, until they add up to a sum of prime numbers, this sum and the number multiplied by the last number are the full number."
" This statement may not sound intuitive, and we can break down this passage with a simple example: starting with the number 1 (the smallest integer), multiply each number after it by 2 (continuously in two times the scale) and get the numbers 1, 2, 4, 8, 16 The sum of these numbers is 1 plus 2 plus 4 plus 8 x 16 x 31 as prime numbers, and according to euclidean's calculation method, the number 496 is a full number by multiplying 31 by the last number in the series, 16.
we can test whether 496 is really a complete number: 496 has a factor of 248, 124, 62, 31, 16, 8, 4, 2, 1, and the sum of these numbers is 1 plus 2 plus 4 plus 8 plus 16 , 31 , 62 , 124 , 248 , 496, which is the third complete number after 6 and 28.
in fact, the first four complete numbers were discovered by Euclidean, 6, 28, 496, and 8128;
Euclid's discovery provides a reliable way to find new full numbers.
but a key problem with this approach is that finding a full number is based on the premise that the sum of the factors is a prime number (as in the previous example, 31), but this is clearly a step that requires a lot of calculation.
Even today, identifying new prime numbers is not an easy task.
modern mathematicians had a more convenient way to think about the results of Euclid, which instead of a lengthy sum, but to rephrase the process of addition to another Greek philosopher associated with complete numbers, Niko Marcus, who lived around 150 AD.
unlike Euclid, NikoMarcus did not write rigorous proof of his findings that his influence on full numbers was not to leave a simple proposition, but to classify numbers.
in his book, Getting Started in Arithmetic, he proposes to divide numbers into excess, loss, and complete numbers.
according to NikoMarcus's classification, a loss refers to the sum of all its factors that are less than the integer of the number itself;
all natural numbers can be grouped into these three types.
Not only does NikoMarcus classify the numbers in this way, he argues that excess and loss are "lower quality" numbers compared to full numbers: "For too many cases, there is excess, excess, exaggeration and abuse;
those that fall between too much and too little, that is, being in equality, produce beauty, beauty, beauty and things like that - the most typical form of which is numbers called complete numbers.
" this classification also influenced many subsequent thinkers who believed that the full number had a sacred nature.
, for example, the 4th-century theologian Saint Augustine wrote in City of God: "Six is a perfect number in itself, not because God created everything in six days;
", more than 1,200 years after the death of Heshimu in Euclid, The Arab physicist and mathematician Heshimu studied the full number of parts of The Geometry original, and became the first person to suggest that the results of Euclid were true.
he believes that the process that Euclidean has used to produce a full number produces an even number.
Heshmu was the first mathematician to try to fully describe all the even numbers, although he could not fully prove the result.
many mathematicians later tried to follow the example of Heshimu's research in the hope of making more progress, but this complete description did not appear until the 18th century. Leonhard Euler, born in 1707
Leonhard Euler, is still one of the most prolific mathematicians.
almost every field of mathematics has a result named after him, and the field of number theory is no exception, and he finally proves that Euclid's algorithm for producing full numbers produces even numbers.
and Euler also found a new full number, a very large number, the eighth full number: however, it was not until the end of the 19th century that the ninth full number was found.
but since then, with the progress of number theory and the advent of the new technology era, mathematicians have finally been able to make public some of the full numbers that are too large to imagine.
51 full numbers so far, mathematicians have found 51 full numbers, the largest full number found in 2018 is more than three million more than the 50th full number found in 2017.
these advances are entirely due to the collaboration between mathematics and computer science.
of course, there is still a big difference between calculating a full number and proving a full number.
While it is now found that a full number is not as difficult as it used to be, there are many questions that need further study, such as what is the common nature of all the full numbers? Are there more than an infinite number of even numbers? Is there an odd number? Source: Principles, Global Science.