When I first started researching about amicable numbers, I was confused on what they were.  Not only that but I felt like I should know what they are because one of the the first websites I opened about them stated that if your are familiar with number theory you should know what amicable numbers are.

After reading more into it and exploring other websites, the explanation of what they were finally clicked. After that I remembered learning about these numbers in class, however I do not recall learning about them prior to my senior year capstone class in college.  Therefore, I would not find it surprising that others do not recall what they are or how they work.

I found through research that there are 42 pairs of amicable numbers below 10^6 with 9 new ones found.  Finding these pairs require a lot of time and precision.  I would not say it is “hard” work to find them once you get the hang of the pattern, but it is most definitely time consuming.

How I understood these numbers and the best way for me to explain this concept is through looking at an example as to how they work.

The simplest form of amicable numbers is the pair (220,284).  To start off we will like the factors of both of those numbers.

220

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

284

1, 2, 4, 71, 142

Now that we see the factors of both numbers, our next step is to add up all of the number in 220 and then again in 284.  When added together, we see the following:

220

1+2+4+5+10+11+20+22+44+55+110=284

284

1+2+4+71+142=220

Since the factors of 220 add up to 284 and the factors of 284 add up to 220, this is therefore an example of a pair of amicable numbers.

A list of more, found from my research is found at my given websites:

Link 1

Link 2

Why am I telling you about these numbers? Well, I was just interested at seeing how many pairs were out there.  Clearly there are more than six, in fact, as said earlier, there are more than 42 pairs which is a lot more than people originally thought.  The first pair, displayed above was said to be originally founded by Pythagoras.  Although there is no yet use for these numbers other than to compare with one another within the pair, when discovered, these pairs did more than just that for the people of that time.

During the time of the discovery, these numbers were said to be magical.  They were used by astrologers to prepare horoscopes.  They were also known to create special ties between two individuals. (I believe that is because of the special tie the numbers had in the pair.)

Back on how many pairs there are.  Even though the research I have found has said that there are around 42-51 pairs of amicable numbers, I have continued my research to find that there are 390 pairs of these numbers.  The people involved in this discovery included the following:

  • Fermat
  • Arab al-Banna
  • Descartes
  • Euler
    • Who found a huge amount
  • Escott
    • offering the other pairs adding up to 390.

So now my research is up to 390 pairs, but I believe there to be much more.  In fact, with infinite being the highest form of a “number” we have, I think that there must also be an infinite number of amicable pairs.  I have no clue how I would solve or prove this, but I am sure it has been or needs to be done.

I liked doing this research.  I feel much more knowledgeable about the subject and found the whole thing very interesting.  I was intrigued by the subject just because of how many unknowns there still are within this subject matter.  I am curious to know if more research is being done and when/if anything more will be discovered.

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