Pythagoras, the pioneer who discovered the perfect number, found that the sum of the true factor 1, 2 and 3 of 6 is also equal to 6.
Plato, an ancient Greek philosopher, put forward the concept of perfect number in the book Republic.
About 300 BC, Euclid, the master of geometry, gave the method of finding the perfect number for the first time in the last proposition of chapter 9 of his masterpiece "Elements of Geometry", which was praised: praise and praise. Euclid theorem: "If 2n- 1 is a prime number, then 2n- 1 (2n- 1) must be a perfect number." And the proof is given.
In the 1 century, Nico Matthews, a member of the Pythagorean school, a famous mathematician in ancient Greece, correctly gave the four perfect numbers 6, 28, 496 and 8 128 in his book Introduction to Arithmetic, and popularly repeated Euclid's theorem of seeking perfect numbers and its proof.
After the mysterious fifth perfect number was born in ancient Greece, it attracted many mathematicians and math lovers to look for it like the gold rush. However, generation after generation has devoted countless efforts, and no one has found the fifth perfect number.
It was not until 1202 that a glimmer of light appeared. Fifi of Italy: fíI. Bonacci, when she was young, traveled with her father to Greece, Egypt, Arabia and other ancient civilizations, and learned a lot of mathematical knowledge. He was brilliant and devoted himself to research and collection after returning to China (s·?·ují): looking for (things) everywhere and collecting them together. Mathematics, wrote the famous book Abacus, became the first person in Europe to spread oriental culture and introduce oriental mathematics to the west in the13rd century, and became a mathematical star on the eve of the Western Renaissance. Fibonacci never let go of the study of perfect numbers. He announced that he had found an effective rule and found the perfect number after calculation. It's a pity that no one sang * * *, and it became a thing of the past.
1460, it was unexpectedly discovered that the fifth perfect number 33550336 was mysteriously given in an anonymous manuscript. This is more than 4000 times larger than the fourth perfect number 8 128. With such a large span, we can imagine the hardships of ancient discoverers by reverse calculation. However, the manuscript does not explain how he got it, nor does it disclose his name, which makes people even more confused: he can't tell right from wrong; Confused, confused. I don't understand.
An Extraordinary Research Course/KLOC-In the 6th century, the Italian mathematician Tattaglia was cut with a knife by the French invaders when he was a child, leaving a stuttering problem, and later became a famous mathematician by self-study. He found that 2n- 1 (2n- 1) is a perfect number when n = 2 and n = 3 to 39 are odd numbers.
/kloc-In a 700-page magnum opus Metaphysics of Numbers, the master of "numerology" in the 0/7th century, Pongers listed 28 so-called "perfect numbers" in one breath. Tattaglia gave 20 pounds, and he added 8 pounds. Unfortunately, neither of them gave proof and calculation process, and later generations found that many of them were wrong.
1963, the mathematician Kaidi finally proved that the fifth perfect number in the anonymous manuscript was correct. At the same time, he also correctly found the sixth and seventh perfect numbers 216 (217-17) and 2 18 (2 19- 1), but he mistakenly thought that 222 (. These three figures were later denied by the great mathematicians Fermat and Euler.
1644, Mei Sen, a French priest and great mathematician, pointed out that only 8 of the 28 "perfect numbers" given by Ponzi were correct, that is, when n = 2, 3, 5, 7, 13, 17, 19, 3.
Without proof, he arbitrarily said: When n ≤ 257, there is only this 1 1 perfect number. This is the famous "Mason conjecture".
"Mason conjecture" has attracted many people's research, and Goldbach thinks it is right; Leibniz, one of the discoverers of calculus, thought it was right. They underestimated the difficulty of perfect numbers.
1730, Euler, known as one of the four greatest mathematicians in the world, was 23 years old and in his prime: absolute beauty was brilliant. Masahiro He gave an excellent theorem: "Every even perfect number is a natural number in the form of 2n- 1 (2n- 1), where n is a prime number and 2n- 1 is also a prime number", and gave a proof that he had never published before. This is the inverse theorem of Euclid's theorem. With the help of two reciprocal theorems of Euclid and Euler, the formula 2n- 1 (2n- 1) becomes a necessary and sufficient condition for judging whether an even number is a perfect number.
After studying Mason's conjecture, Euler pointed out: I dare to assert that every prime number less than 50 or even less than 100 makes 2n- 1 (2n- 1) a perfect number, and only n takes 3, 5, 7, 13,/kloc-. I got these results from a beautiful theorem, and I have great confidence in them. "1772, Euler lost his sight because he worked too hard, but he still didn't stop studying. In a letter to the Swiss mathematician Daniel, he said, "I have proved mentally that n = 31:220 (231-1) is the eighth perfect number." At the same time, he found that his idea that n = 4 1 and n = 47 are perfect numbers is wrong.
Euler theorem and the eighth method of the perfect number he discovered. The study of perfect numbers has undergone profound changes, but people are still not thorough (chè d ǐ): all the way to the end, thorough and thorough, but also thorough. Solve the "Mason conjecture".
1876, French mathematician Lucas discovered a new method to test prime numbers, and proved that n = 127 is indeed a perfect number, which made one of the "Mason conjecture" come true. Lucas' new method brings a glimmer of life to those who study the perfect number, but it also shakes the "Mason conjecture". Through his method, the factorizer found that n = 67 and n = 257 were not perfect numbers.
In the next 48 years from 1883 to 193 1 year, mathematicians found that in the range of n ≤ 257, three perfect numbers when n = 6 1, 89 and 107 were omitted.
At this point, people wave upon wave, constantly exploring new paths and creating new methods. It took more than 2000 years to calculate paper records with a pen, and * * found 12 perfect numbers, which were n = 2, 3, 5, 7, 13, 17, 19, 3 1 6/respectively.
Descartes once publicly predicted: "There are not many perfect numbers, just like human beings, it is not easy to find a perfect person."
History has proved his prediction.
Since 1992, people have found perfect numbers with the help of high-performance computers. In 1996, only 18 were found.
Waiting to debunk the mystery: q √. Up to now, all the 30 perfect numbers found are even numbers, so mathematicians put forward a guess (cā icè): guess, which is estimated by imagination. There is no odd perfect number.
1633 1 1 month, the French mathematician Descartes wrote a letter to Mei Sen, which initiated the study of odd perfect numbers for the first time. He thinks that every odd perfect number must have the form of PQ2, where P is a prime number, and claims that he will find it soon, but until his death, and so far, no mathematician has found an odd perfect number. This has become another big problem in the world number theory.
Although no one knows whether they exist, one thing is clear after generations of mathematicians' research and calculation. In other words, if there is an odd perfect number, it must be very large.
How big is it? Far from saying, the investigation of the contemporary great mathematician Orr (jiā nchá): Look carefully to find problems; Turn over the examination paper. Want a natural number below 10 18, there is no odd perfect number; 1967, Tackman announced that if the odd perfect number exists, it must be greater than 1036, which is a 37-digit number; 1972 must be greater than 1050, 1982 must be greater than10/20; ..... This elusive odd perfect number may exist, but it is too big for people to calculate with computers.
It is also an anecdote in mathematics to produce so many estimates about the existence of odd perfect numbers!
There are still many mysteries to be solved about perfect numbers, such as: What is the relationship between perfect numbers? Is the perfect number finite or infinite! Do you have odd perfect numbers? People have also discovered the wonderful phenomenon of perfect numbers. Add up all the digits of a perfect number to get a number, and then add up all the digits of this number to get another number. If you keep doing this, the result must be 1. For example, for 28,2+8 =10, for 496, 1+0 = 1, 4+9+6. 19, 1+9 = 10, 1+0 = 1 and so on. Does this phenomenon hold true for all perfect numbers except 6? These problems, like other math problems, need to be overcome (?n?kè: capture (enemy strongholds)). .