As I've started reading more about pure and discrete mathematics I've encountered a very interesting problem and thus I thought it would be just an interesting quiz for this weekend.
The quiz of the day
What is the number that matches the "?" symbol?
1. |
220 |
284 |
2. |
1184 |
1210 |
3. |
2620 |
2924 |
4. |
5020 |
? |
Have you found the answer? Not sure?
The answer is...
5564. Hopefully you've found the same answer. If you did then you already know what an amicable number is. If not then let me show you what Wikipedia says about this:
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3 but not 6.)
The (proper) divisors of 220 are: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 and their sum is 1+2+4+5+10+11+20+22+44+55+110=284
The (proper) divisors of 1184 are: 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 and their sum is 1+2+4+8+16+32+37+74+148+296+592=1210
The (proper) divisors of 2620 are: 1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310 and their sum is 1+2+4+5+10+20+131+262+524+655+1310=2924
The (proper) divisors of 5020 are: 1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510 and their sum is 1+2+4+5+10+20+251+502+1004+1255+2510=5564
These numbers are known even from the time of Pythagoreans, ca. 300 BCE (they knew only the first pair; it is unknown how occurred them this problem but it did). Anyway somewhere in the 9th century a great Iraqi mathematician named Thabit ibn Qurra found a rule that helped him finding two more pairs. So far three pairs were known. Somehow, perhaps because there was no Internet at that time, the Thabit's "discovery" wasn't known (yet) in Europe. In 1636 a French mathematician named Pierre de Fermat had found a new pair (namely 17296;18416). Another French mathematician that was contemporary with Fermat was Rene Descartes. These two guys were rivals and thus they just hated each other. When Descartes found out that Fermat had found a solution to that problem he became frustrated and probably hadn't slept well until he came up with a new pair: 9363584 and 9437056. So far only three different pairs of amicable numbers were known/discovered. One hundred years later the greatest mathematician of all the times, Leonhard Euler, had wrote a paper called "De numeris amicabilibus" where he explained a general rule to find such numbers. He found 58 new pairs! That's interesting because what all the mathematicians together have found in 2000 years he found 20 times more in only a short period of time. That great was Euler! (although this wasn't his greatest contribution; no sir, not at all! his contribution to mathematics, physics, mechanics and even music is much far than that). Keep in mind that as of 1946 only 390 pairs were know and 15% were found only by Euler, so that tells something about Euler, right?. According to Wikipedia "In 2007, there were almost 12,000,000 known amicable pairs". This was probably possible thanks to the todays computer aid.
Anyway, I find very fascinating these numbers although they don't have a real-life applicability. If you want to see something really extraordinary then check this out:
which for 
becomes 
. This is the most beautiful equation of all times and it's known as Eulers's identity.
Why is beautiful and interesting you might ask. Let's see:
- it contains the top 5 most magic numbers:
=3.1415.., e=2.7182.., i where i2=-1, 0(the additive identity), 1 (the multiplicative identity) - it contains the most known arithmetical operations: addition, multiplication, rise to a power and even subtraction if you rewrite little bit the equation
You cannot refrain asking yourself how did the Euler come to this equation. The only answer is he was a genuine genius. Euler's collected work runs to over 25000 pages (he died at age 76; 25000/76 means ca. 1 page/day starting with his birth).
Now, if you think that this article was interesting don't forget to rate it. It shows me that you care and thus I will continue write about these things.
Eugen Mihailescu
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