A self (it's like many popular warnings, a wonderful laundry list, in which many favorite things can be put).
There are also friends in the number, 220 and 284, each of which is equal to the sum of each other's divisors. (The divisible number of 220 is: 1.
The divisible number of 245101110,248 is: 1 2 7 1 142.
220). The number of another pair of friends is 1636 more than two thousand years later, written by Fermat (Pierre de).
Fei Erman): 17296 and 184 16. Later mathematicians discovered more. Worth mentioning
It is the 60th and smallest pair, 1 184 and 12 10, which was changed by a 16-year-old in 1866.
Italian students found it. Pythagoras prefers perfect numbers, that is, the sum of all its divisible numbers is itself. Like 6.
The divisibility of is 1 2 3, the divisibility of its sum 6 and 28 is 1 2 4 7 14, and its sum is 28. Divide by 6 and then
In addition, the ancient Greeks also knew 496 and 8 128, and the fifth perfect number 33550336 was developed more than 700 years later.
Now By April of 1998, * * * found 37 perfect numbers, all even numbers.
17th century, the famous French mathematician Fermat (p Fermat, 1608- 1665) got a theorem named after him: If n is a prime number and A is an arbitrary natural number, then a2-a is a multiple of n, is the inverse proposition of the above theorem valid? After Fermat, there are countless researchers. The famous German mathematician Leibniz (1646- 17 16) once suggested that if n is not a prime number, then 2n-2 is not a multiple of n. So in Leibniz's view, when a=-2, the inverse proposition of Fermat's Last Theorem holds: If a-a is a multiple of n, Coincidentally, Li (1811-1882), a great mathematician in Qing dynasty, summed up a method of judging prime numbers in 1869 (Li called it "number root"): multiply the known number by the logarithm of 2, and then multiply the product. If the remainder is divisible by the known number, the known number is a prime number; Otherwise, it is not a prime number. The above method is simply: let n be a known natural number, if 2 n-2 is a multiple of n, then n is a prime number, otherwise n is not a prime number. The counterexample found by mathematicians completely negates Leibniz's and Li's conclusion that although 234 1-2 is a multiple of 34 1, 341=1is a combination. Later, people found more counterexamples: 56 1, 645, 1 105, 1387, 1729, 1905, 2407, ...
This introduction of mathematicians' mistakes and the twists and turns of mathematical development in history can change students' wrong views on mistakes. Let them understand that the "nobility" of mathematics is only a cultural activity of human beings, and any study and research will encounter mistakes, setbacks and failures. Therefore, it is an effective method to change students' wrong views with the history of mathematics.
Fermat number conjecture: the master's mistake
1640, Fermat, who left an indelible footprint in the field of number theory, thought about a question: whether the value of formula 22n+ 1 must be a prime number. When n takes 0, 1, 2, 3 and 4, the corresponding values of this formula are 3, 5, 17, 257 and 65537 respectively. Fermat found that all five numbers are prime numbers. Therefore, Fermat put forward a conjecture: a number with the shape of 22n+ 1 must be a prime number. In a letter to a friend, Fermat wrote: "I found that a number in the form of 22n+ 1 is always a prime number. I pointed out to analysts a long time ago that this conclusion is correct. " Fermat also admitted that he could not find a complete proof.
The number 22n+ 1 studied by Fermat has a wonderful form, and it was later called fermat number, denoted by Fn. Fermat's guess at that time was equivalent to saying that all fermat number must be prime numbers. Is Fermat right?
It is not easy to further verify Fermat's conjecture. Because with the increase of n, f n increases rapidly. For example, F5 =4294967297 needs to be tested first, which is already ten digits for future generations. Probably, because this number is too large, Fermat did not verify his guess. So, is it a prime number as Fermat thinks?
1 729 65438+February1,Goldbach (the proposer of Goldbach conjecture) asked in a letter to Euler: "Fermat thinks that all numbers with the shape of 22n+ 1 are prime numbers. Do you know this problem? He said he couldn't prove it. As far as I know, no one else has proved this problem. "
This question attracted Euler. 1732, Euler, who was only 25 years old, got F5 = 641× 670041= 5× 27+167 years after Fermat's death, so Fermat's guess was wrong.
In fermat number's research, Fermat, the great genius of number theory, paid too much attention to his intuition and rashly made the only wrong guess in his life. More unfortunately, the progress of research shows that Fermat is not only wrong, but also very likely wrong.
Since then, people have studied more about fermat number. With the development of electronic computers, computers have become a powerful tool for mathematicians to study fermat number. Even so, a Fermat prime number has not been added to the known fermat number. So far, Fermat prime numbers have not been found except for the five confirmed by Fermat himself! Therefore, people began to suspect that all fermat number except the first five are composite numbers. If this conclusion is confirmed, I'm afraid Fermat's hasty guess will not have a worse ending.