Amicable Numbers

     Before this blog I had no idea what Amicable numbers were so I had to go out and learn about them and this is what I have learned. Amicable numbers are two integers greater than or equal to 1 whose sum of  proper positive divisors (any divisor of some positive integer n other than n) of one of the integers is equal to the other integer, and vice versa. A pair of amicable numbers form an aliquot sequence (a recursive sequence in which each term is the sum of the proper divisors of the previous term) of period 2. 

Amicable numbers were known by Pythagoreans who thought they had mystical properties. Amicable numbers are sometimes called friendly numbers, the story behind that is included on shyamsundergupta website in the amicable numbers section were it is said to be when Pythagoras was asked "what is a friend?" and replied that a friend is one "who is the other I, such as 220 and 284".  (220,284) are the smallest and most well know pair of amicable numbers. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which if we sum together we get 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which if we sum together we get 220.
     A general formula has been derived from Iraqi mathematician Thābit ibn Qurra, the method for discovering amicable numbers states that if

p = 3 × 2^(n − 1 )− 1,

q = 3 × 2^n − 1,

r = 9 × 2^(2n − 1) − 1,

where n > 1 is an integer and p,q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers. So if we choose are number to be 2 we get p = 3 x 2^((2) - 1) -1 = 5, q = 3 x 2^2 -1 = 11, and r = 9 x 2^(2(2)-1)-1 = 71. We can see that are p, q, and r are all prime numbers  and n > 1 so we have satisfied the hypothesis which means that 2n×p×q and 2n×r are a pair of amicable numbers so if we plug in our values we got for p, q, and r we get 2(2)×5×11 = 220 and 2(2)×(71) = 284 which as we know are amicable numbers. If you play around with this formula trying new n values like I did you will quickly realize it is not often you get p,q,and r to all be prime numbers. This method although is not perfect because it cannot generate co-prime amicable numbers but it is still unknown if a co-prime amicable number pair even exist.
     Euler also took an interest in amicable numbers, he found 59 new pairs of amicable numbers. Euler generalized Thābit's Rule in Euler's rule for amicable numbers. In Euler's rule it states let n be a positive number, and choose 0< x < n such that g = 2^(n-x) + 1. If
p = 2^x × g - 1
q = 2^n × g - 1
s = 2^(n+x) × g^2 - 1
are all prime numbers then
 (M,N) = (2^n × p × q, 2^n × s)
is an amicable pair. We can see that Euler made some slight modifications to Thābit's original conjecture and because of it was able to find new pairs of amicable numbers. Although Euler's modifications to the formula still could not fix the issue of the formula not being able to generate co-prime amicable numbers.

     Like a handful of topics in mathematics there are many things we have yet to figure out or prove. There are still many things we do not know about amicable numbers, for example in every know case, the pair of amicable numbers are either both even or both odd numbers. It is unknown if an even-odd pair of amicable numbers exist. I mentioned earlier that it is still not known if a pair of coprime amicable numbers exist since every known pair shares at least one common factor higher than 1. Using Thabit's formula or any similar formula (Euler's Rule) it is not possible to generate a pair of coprime amicable numbers. It will be interesting to see if someone finds a new method for calculating amicable numbers that is better than Euler's or Thabit's formula and if any of these special case pairs are ever discovered. 

How Greek Mathematicians Influenced Modern Mathematics

Math as we know it today has been influenced by so many people but one of the biggest contributors were the Greek mathematicians. The Greeks have heavily influenced modern mathematics in a variety of ways many being in geometry but I am only going to focus on one of their contributions which I would arguably say was their greatest contribution. In my opinion the biggest way Greek mathematicians influenced modern mathematics was with the idea of proofs. Mathematical proofs have now become a common occurrence in math, anyone studying math has had to learn how to write mathematical proofs. The reason why I believe that the Greek mathematician's greatest influence was the idea of the proof because proofs changed the way people thought about mathematics. They were now looking at math in a more abstract way and focused more on finding patterns that could be applied to multiple cases and less on solving individual problems. The ancient Greeks used proofs to show that math wasn't just about performing calculations but math could actually be used to understand the world around us. Instead of focusing on literal values they generalized things so they could find universal patterns in mathematics. They went about this with proofs which were convincing arguments that showed that something was true. The early Greek math proofs were mainly all geometric demonstrations, such as Pythagoras's proof of the Pythagorean Theorem. In this proof Pythagoras showed that the area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the remaining squares. In mathematics today geometric demonstrations are not a very popular form of proof. It was the Greek Mathematician Euclid who changed how things were proved by establishing definitions and postulates which he used to proof his propositions. The proofs and definitions Euclid established formed the basis of mathematics. His style of proving things is being used and taught today to prove new conjectures in math. Greek Mathematicians didn't just influence modern math but they influenced the way we understand, view, and verify our new discoveries in math.  

What is Math?

One question I just never know quite how to answer is, what is math? Math is such a broad topic that encapsulates so many ideas and theories, some to which I can not even begin to comprehend, that it's hard to answer that question in just a few lines. I think Math in it's most basic form is about pattern recognition. Often in math when we start to learn a new idea or theorem we keep using it again and again with different numbers or different equations and look to see what patterns we can find. From that pattern recognition we can start to propose conjectures which we can later prove to form theorems. From these theorems we can use them to prove new theorems and help math constantly evolve discovering new concepts and ideas. That being said I think one of the most important moments in math was the creation of a number system because it allowed people to quantify things and really begin to tackle math problems in a conceptual way. Other important moment/discoveries in math would be some of the first math proofs for example the geometry proofs by the Greek mathematicians such as the famous Pythagorean theorem. I think some of these early geometry proofs were key in mathematics because they provided a basis for mathematics for which other mathematicians could use and build off of to advance math even further. As we continue to advance math more and more each day I wonder if the question what is math will get harder and harder to answer.