I've recently read a book called Visions of Infinity by Ian Stewart. Stewart provides a nice overview of all the most formidable and important problems mathematicians have solved throughout time and problems that have yet to be solved yet. The flow of Visions of Infinity was really interesting because each chapter, excluding the first and last chapter, looks at a different famous problem and is independent from each other chapter. So each chapter looks at a famous problem that someone proposed to true or unsure if it was true or false, then Stewart gives us a glimpse into how from once the problem was first brought up the different conjectures made for it all the way to either how the problem was solved or how far the problem was been taken and what is potentially needed to solve them or if they may ever be solved such as the Riemann Hypothesis and P/NP problem. One of the main points that I felt Stewart was trying to show in each chapter was that these famous problems aren't famous so much because of how useful the knowledge we get from solving them is but instead all the great things we learned while trying to solve these famous problems that may have never come up without people trying to solve these famous problems such as the Taniyama-Shimura conjecture about elliptic curves that was proven while trying to solve Fermat's Last Theorem.
   One of my favorite chapters in the book was the chapter dealing with the four color theorem. The reason I really enjoyed this chapter was because it gave a lot of background and insight about the four color theorem along with why it was so important. I found it very interesting that the problem was first proposed outside a mathematical setting by Francis Guthrie while he was coloring the counties in a map of England. When I first learned about the four color theorem in my Discrete Math class I thought that the importance of the proof was just proving the theorem itself. I learned from Visions of Infinity that the importance of the four color theorem was that it changed what would be accepted as proofs. Before the four color theorem proofs use to rely solely on human brainpower so when the theorem was proven with computer assistance many mathematicians were hesitate to accept the proof. I agree with the author when he said the proof was not radically altered if some of the steps were done by a computer. My favorite quote from the author is one regarding computer assisted proofs is, "a proof is a story; a computer-assisted proof is a story that's too long to be told in full, so you have to settle for the executive summary and a huge automated appendix."
   Overall I really enjoyed Visions of Infinity, one thing I really liked about the book is that it showed me how math theorems and proofs in real life are much more of a team effort. Through my undergraduate career it seems like all the proofs I write for my math classes have to be done on my own and no collaboration with anyone else, so I found it very interesting when the author wrote about other mathematicians getting stuck on theorems and reaching out to other mathematicians to help them. Another thing I really liked about the book was the author pushing the idea that some of the most important proofs of all time were not because of the things they proved but because of the questions that arose from trying to solve them and the important things we proved. A good example being the Taniyama-Shimura conjecture because the author mentioned no one ever really thought about elliptic curves until mathematicians tried to prove Fermat's last theorem. One downside to the book is that to really understand a lot of these proofs they require a lot of knowledge about all different areas of math but it is understandable that the author expected the reader to possess quite a bit of math knowledge before reading the book because if he tried to teach the reader all the topology, number theory, and other math areas needed to understand the thinking and proofs the book would be exponentially longer. That being said I would still recommend this book to anyone interested in mathematics because there is still lots of information about logic, proofs, and mathematics to take away from this book.

Is Math A Science?

     To compare math to a science we need to first clearly define what a science is. An interesting definition I found online was from Explorable which defined science as "the observation, identification, description, experimental investigation, and theoretical explanation of phenomena." It is in particular used in activities applied to an object of inquiry or study. Looking at that definition we can see that math does have some of these traits that define a science for example before we come up with any kind of conjecture in math we first observe and experiment to identify patterns so that we can formulate the most accurate conjecture. I think math perfectly fits the part in the definition about science being a theoretical explanation of phenomena which is why I would argue for math being a science. My best example for math being a theoretical explanation of phenomena would be discrete dynamical systems, in my independent study on discrete dynamical systems we looked at how different factors affected population numbers. The University of Texas has a cool article talking about a professor using  discrete dynamical systems to model natural phenomena. So if we define science the way that Explorable does I would say math is indeed a science.
     Although using other definitions for science such as Webster's definition which is that science is "the knowledge about or study of the natural world based on facts learned through experiments and observation." I would disagree that math is a science with this definition because math may study the natural world in some cases but the facts are not based on experiments and observation but rather on proof. We may use empirical evidence to help with formulating conjunctures and disproving them but for us to accept them as fact we need to generalize them and prove they still hold. Another important difference between math and this definition is that math is not just knowledge about the natural world one easy example would be in linear algebra when we would have vectors that were in the 4th dimension and higher. In math mathematicians can create strange worlds using many different areas of math and since the definition we are using here limits the knowledge to just about the natural world I would say math is not science if we define science in this way.  
     So it really depends on how you define science for whether or not math is a science. Personally the definition of science I like the best is comes from Dictionary.com which defined science as " a branch of knowledge or study dealing with a body of facts or truth systematically arranged and showing the operation of general laws" coming from the online dictionary. I think that this definition really captures the essence of what science is which is basically an ordered collection of information showing how things work. So to me math is a science because it is a system of knowledge about numbers, shapes, and abstract ideas which are built off sets of axioms, which can be thought of as basic rules or general laws something must follow. I believe the question of whether math is a science is a tough question because there exist many different definitions for what a science is, and depending on what definition you pick can affect your answer.

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.