Amicable numbers are two different numbers so related that the sum of the proper divisors (positive factor of the number other than the number itself) of each is equal to the other. Amicable numbers were found by the Pythagoreans, who credited them with many mystical properties. However, many mathematicians over the years have contributed to the knowledge and examples we have of amicable numbers. The amicable pair (17,296 ; 18,416) is often attributed to Fermat but was actually discovered by the Arab al-Banna in the late thirteenth or fourteenth century, which seems to be the first account of a discovered pair. The pair consisting of 9,363,584 and 9,437,056 was found by Descartes. A remarkable number of pairs of amicable numbers, 59 in all, were found by Euler. Quite some time after Euler's discovery, a sixteen year old Italian youth found a smaller, overlooked pair of amicable numbers, 1184 and 1210. Some time after these small discoveries, E. Escott wrote a long paper dedicated to the amicable numbers, which offered an inventory of 390 amicable pairs. Each of these mathematicians contributed to this ongoing investigation. I say that it is an ongoing investigation because of the unanswered questions that we have. Mathematicians today still investigate and work with amicable numbers hoping to discover something new and advanced.

There are still some unanswered questions that we have about amicable numbers. It is unknown where there exists infinitely many pairs of amicable numbers. We know that there exist pairs of odd amicable numbers, but it is not known whether there exists any pair with one of the numbers odd and one even. It is not known whether there are pairs of relatively prime amicable numbers. However, H. J. Kanold proved that if there existed a pair of relatively prime amicable numbers, then each of the numbers had to be greater than 1023 and they would have together more than 20 prime factors. There are always new and advanced discoveries appearing in mathematics, so maybe someday these unknowns will no longer occur.

I have not done that much work with amicable numbers, personally, so I thought it would be ideal to research the idea and process behind them. It seems to be very complicated to find these amicable numbers because of the tedious work. It is much easier and likely to find numbers that are not amicable. I believe that an easy way to look at amicable numbers would be to first avoid factors of that number because it will obviously be much larger than that. Another more simple way would be to start with a number and see what the sum of the factors is, then check to see if it works both ways. These seem to be the most simple solution to finding a pair of amicable numbers or check to see if a pair are amicable. Other than that, I do not believe there is a system that seems to always work for amicable numbers. They are still quite a mystery to us, but we are learning and discovering more every day.

Example worked out:

220: 1+2+4+5+10+11+20+22+44+55+110=

284: 1+2+4+71+142=

YouTube Video - Numberphile

There are still some unanswered questions that we have about amicable numbers. It is unknown where there exists infinitely many pairs of amicable numbers. We know that there exist pairs of odd amicable numbers, but it is not known whether there exists any pair with one of the numbers odd and one even. It is not known whether there are pairs of relatively prime amicable numbers. However, H. J. Kanold proved that if there existed a pair of relatively prime amicable numbers, then each of the numbers had to be greater than 1023 and they would have together more than 20 prime factors. There are always new and advanced discoveries appearing in mathematics, so maybe someday these unknowns will no longer occur.

I have not done that much work with amicable numbers, personally, so I thought it would be ideal to research the idea and process behind them. It seems to be very complicated to find these amicable numbers because of the tedious work. It is much easier and likely to find numbers that are not amicable. I believe that an easy way to look at amicable numbers would be to first avoid factors of that number because it will obviously be much larger than that. Another more simple way would be to start with a number and see what the sum of the factors is, then check to see if it works both ways. These seem to be the most simple solution to finding a pair of amicable numbers or check to see if a pair are amicable. Other than that, I do not believe there is a system that seems to always work for amicable numbers. They are still quite a mystery to us, but we are learning and discovering more every day.

Example worked out:

220: 1+2+4+5+10+11+20+22+44+55+110=

__284__284: 1+2+4+71+142=

**220****Examples from 1 to 20,000,000**

YouTube Video - Numberphile