Numbers or numbers?

Affinity number, also known as blind date number, friendly number and friendly number, means that the sum of all positive integer factors of two positive integers (except themselves) is equal to the other. Pythagoras once said, "A friend is a beautiful image of your soul. Be close to 220 and 284. "

You can also buy a two-half heart-shaped key chain or jewelry engraved with "220" and "284" respectively. People buy them, give half to their loved ones and keep the other half for themselves. I did the same thing. According to legend, in ancient Greece, 220 and 284 were symbols of friendship and romance. Until now, some nerds still use this meaning.

220 and 284:

All positive factors of 220 (excluding itself) add up to1+2+4+5+10+1+20+22+44+55+1.

The sum of all positive factors of 284 (excluding itself) is:1+2+4+71+142 = 220.

The factors of 220 include 1, 2,4,5, 10,1,20,22,44,55 and 1 1 0. They don't look strange, but if you add them up, you will find that their sum is exactly equal to 284. Nothing special, okay? Then add up all the factors of 284 (1, 2,4,71,142), and the result is -220. Add all the factors of one number and you will get another number. 220 and 284 are so closely related that you get a name: "affinity number" (blind date number).

These two numbers are not the only kinship numbers. Fermat found a pair of new affinity numbers in 1636, which are17,296 and18,416 respectively. But to use them, you may have to buy a bigger key chain or jewelry. Rene descartes found a pair of affinity numbers in 1638-9363584 and 9437056. If you want to use these two numbers, it is estimated that you can only be marginal. 1747, Euler also joined the game of finding affinity numbers, found about 60 pairs of new affinity numbers, and showed off well. But no one has found the second smallest affinity pair-1, 184 and1,2 10. These three affinity pairs were adopted by B. Nicolo i., a middle school student who was only 16 years old in 1866. We have no way to prove whether the motivation he found came from love or studying mathematics.

If you want to know more about the affinity number, you can find all the information about the known affinity number and its discoverer on the following website (friendly. home page. dk).

We still know very little about affinity numbers. There is a long-term conjecture that all affinity numbers are multiples of 2 or 3, but in 1988-42, 262, 694, 537, 5 14, 864, 075, 544, 955, 198, 66. 188,606,697,466,971,84 1, 875 proved that this conjecture was wrong. So the guess becomes again: all affinity numbers are multiples of 2, 3 or 5, but in 1997, people find another counterexample that contains 193 numbers. There are also conjectures that there are infinite pairs of affinity numbers, but even if at least 1 1, 994,387 pairs of affinity numbers are found, to be honest, I don't know who to trust.

12,496 is the change of affinity number. Add up its factors and it is 14288. Add up the factors of14,288 to get15,472; If this process continues,15,472 will become14,536,14,536 will become14,264,14,264 will become 12. But anyway, this trip is really exciting! By summing these factors, we get this cycle consisting of five numbers. This number chain is called "social number". There are also communication signals with a cycle length far exceeding 5. Although they are not as close as affinity numbers, we are open to them. You may have noticed that we exclude the original number itself from the factors and sum the so-called "appropriate factors" (that is, all factors including 1 but excluding the original number itself). ]

Next, the most magical number comes: there are some rare numbers, and when you add up their factors, you will get the original number. The smallest example is that the factor of 6 is 1, 2 plus 3,1+2+3 = 6; Then it is 28, because 28= 1+2+4+7+ 14. The ancient Greeks called these numbers "perfect numbers". The next perfect number is 496, and then there is a big leap to 8 128. After that, it became more and more ridiculous. The next perfect number is 33,550,336, followed by 8,589,869,056, followed by 65,438+037,438, 6965,438+0,328, followed by 2,305,843,008,658.

The ancient Greeks discovered the first four perfect numbers within 8 128. 33,550,336 was recognized as a perfect number for the first time in 1456, and the next seven perfect numbers were discovered one after another in the next 500 years. The largest perfect number contains 77 numbers. Since 1952, the application of computer has found 36 other larger perfect numbers. The largest known perfect number is found in 20 13, which contains 34,850,340 numbers (the last number is 6). This is very shocking. Its true factor is as many as115,770,321,which adds up to itself.

The discovery of the perfect number is closely related to a problem we have already mentioned: "Looking for mersenne prime". So far, all the perfect numbers found are multiples of mersenne prime. In Euclid's time, he defined perfect numbers as "the sum of fractions" in the seventh volume of Elements of Geometry, and proved that all even perfect numbers have a mersenne prime factor. Euler later proved a (slightly different) conclusion: all mersenne prime is a factor of perfect number (Euclid and Euler finally succeeded in combining, and this is not only because they are both surnamed "Ou"). So whenever we find a mersenne prime, we will get a perfect number for free at the same time.

There is another missing aspect of perfect numbers: odd perfect numbers. So far, all we have found are even perfect numbers, but odd perfect numbers are completely possible. If they exist, we know that there is no mersenne prime in their factors, and they will have some properties that we never thought of. Although most people guess that there is no odd perfect number, the search for odd perfect numbers has never stopped. This requires a lot of computing resources, so naturally, a distributed computing project is looking for them. If you want to join, you can log on to oddperfect.org to find out.

The above excerpt is from what can we do in four-dimensional space newly published by Houlang, and [Meet] has been authorized.