Analyzing the Goldbach Conjecture
Introduction
Proposed in 1742 by Christian Goldbach, the Goldbach Conjectures have remained among the most famous mathematical problems ever proposed. The conjectures were first identified in a letter by him written to fellow famous mathematician, Leonhard Euler, the founder of Euler’s constant (2.718). The conjectures basically state that every even integer greater than two is a sum of two primes, and every odd integer greater than five is a sum of three primes. Though there is still a lot to discover about the conjectures, mathematicians generally view them as true. In this mathematical exploration I shall discuss in detail both of Goldbach’s conjectures and the mathematical procedures that occur.
Rationale
The reason I chose to research this topic is because a while back I had realized that every even number other than two could be reached by adding two prime numbers. I was in Algebra class and I was adding numbers together and I noticed a pattern emerging that the even numbers were always the sum of two odd numbers. Moreover when adding more numbers together I realized it was actually the sum of two prime numbers. My Algebra teacher told me she never noticed this pattern and this fascinated me as I thought had found something that no one had ever found before. When researching possible Calculus Exploration IA topics I came upon this topic again. That is why I chose to do this project.
Aim of Exploration
For over 270 years, mathematicians have been analyzing the Goldbach Conjecture. In all this time no one has been able to completely prove the strong conjecture. The weak conjecture was proved using the Riemann hypothesis. The aim of this exploration is to determine the validity of this co...
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...in no time they will find a formula that will be able to prove the whole theorem..
Reflection
Overall, I enjoyed researching this topic and learned a lot from it. When i first saw this topic, I assumed that there was nothing to it and it was quite simple. However, when I started researching the topic I found out how mistaken I was. The accumulation of evidence leading to the proof of the Weak Conjecture and the information on the Strong Conjecture was substantial. It was quite interesting seeing how a theorem, that at first glance seemed so simple was actually quite complicated. It took over 270 years just to solve the weak conjecture! My view of mathematics and calculus has changed as I now see that what we are doing currently is nothing compared to the difficulty of these equations. It was an enjoyable experience and I would not mind writing another exploration.
Goldbach’s conjecture is one of the most well-known theories in all of mathematics. His conjecture states that, “every even integer greater than 2 can be expressed as the sum of two primes.” Goldbach’s conjecture includes the Goldbach number and many other algebraic expressions. Goldbach’s conjecture is so crucial that it was even featured in Hans Magnus Enzensberger’s The Number Devil. During the 5th night, the number devil shows Robert the Goldbach conjecture. On page 98 of The Number Devil, the number devil gives Robert examples of how to solve and work Goldbach’s conjecture. The number devil uses triangles as an example to introduce Goldbach’s conjecture. The number devil makes Robert throw coconuts to make triangles. This example shows a perfect example of Goldbach’s conjecture because it shows that “every even integer greater than 2 can be expressed as the sum of two primes.” The number
On the outset of the acceptance of this idea, one will find that the number of
...mselves. It is this lack of an external check that makes it very difficult to construct a proof wrought from pure reason that is neither circular nor falsely assuming. In science, checks our found in phenomenon. If a theory is logically sound but does not work in the physical world, it is ruled out. Maybe we will find a similar check for ideas, or maybe we will devise a way around this problem of checking ideas. Either way, the problem is present, and it seems that ideas are not a likely place to find truth.
This book would be used as an extension to our lesson on odd and even numbers by supplementing it with illustrated examples and delivering the material through a meaningful story. Our lesson went over exactly what odd and even numbers were and how to identify them. The main example provided in the text suggested calling students up three students to the front of the class. The students were asked if each child could be paired up with a partner, obviously the answer was no because one student would be considered the “odd man out.” The teacher then called up a fourth student and asked the same question, this time they could
Fermat’s Last Theorem--which states that an + bn = cn is untrue for any circumstance in which a, b, c are not three positive integers and n is an integer greater than two—has long resided with the collection of other seemingly impossible proofs. Such a characterization seems distant and ill-informed, seeing as today’s smartphones and gadgets have far surpassed the computing capabilities of even the most powerful computers some decades ago. This renaissance of technology has not, however, eased this process by any means. By remembering the concept of infinite numbers, it quickly becomes apparent that even if a computer tests the first ten million numbers, there would still be an infinite number of numbers left untested, ultimately resulting in the futility of this attempt. The only way to solve this mathematic impossibility, therefore, would be to create a mathematic proof by applying the work of previous mathematicians and scholars.
...trass saw these problems form a purely mathematical point of view and that helped them redefine the mathematical concept of a limit. Others have thought of these paradoxes as a way of feeding our skepticism and doubting the deficiency of what we presume.
In 1629, a Flemish mathematician, Albert Girard, published a book called L’invention nouvelle en l’ Algebre. In his book, he claimed that there were always n solutions for equations of degree n. However he did not assert that solutions are of the form a + bi, w...
(222). This all lead to Descartes coming up with a theory that “perhaps we do not even
Pierre de Fermat Pierre de Fermat was born in the year 1601 in Beaumont-de-Lomages, France. Mr. Fermat's education began in 1631. He was home schooled. Mr. Fermat was a single man through his life. Pierre de Fermat, like many mathematicians of the early 17th century, found solutions to the four major problems that created a form of math called calculus. Before Sir Isaac Newton was even born, Fermat found a method for finding the tangent to a curve. He tried different ways in math to improve the system. This was his occupation. Mr. Fermat was a good scholar, and amused himself by restoring the work of Apollonius on plane loci. Mr. Fermat published only a few papers in his lifetime and gave no systematic exposition of his methods. He had a habit of scribbling notes in the margins of books or in letters rather than publishing them. He was modest because he thought if he published his theorems the people would not believe them. He did not seem to have the intention to publish his papers. It is probable that he revised his notes as the occasion required. His published works represent the final form of his research, and therefore cannot be dated earlier than 1660. Mr. Pierre de Fermat discovered many things in his lifetime. Some things that he did include: -If p is a prime and a is a prime to p then ap-1-1 is divisible by p, that is, ap-1-1=0 (mod p). The proof of this, first given by Euler, was known quite well. A more general theorem is that a0-(n)-1=0 (mod n), where a is prime...
Carl Friedrich Gauss was born April 30, 1777 in Brunswick, Germany to a stern father and a loving mother. At a young age, his mother sensed how intelligent her son was and insisted on sending him to school to develop even though his dad displayed much resistance to the idea. The first test of Gauss’ brilliance was at age ten in his arithmetic class when the teacher asked the students to find the sum of all whole numbers 1 to 100. In his mind, Gauss was able to connect that 1+100=101, 2+99=101, and so on, deducing that all 50 pairs of numbers would equal 101. By this logic all Gauss had to do was multiply 50 by 101 and get his answer of 5,050. Gauss was bound to the mathematics field when at the age of 14, Gauss met the Duke of Brunswick. The duke was so astounded by Gauss’ photographic memory that he financially supported him through his studies at Caroline College and other universities afterwards. A major feat that Gauss had while he was enrolled college helped him decide that he wanted to focus on studying mathematics as opposed to languages. Besides his life of math, Gauss also had six children, three with Johanna Osthoff and three with his first deceased wife’s best fri...
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
Mathematics is an area of knowledge where the claim is applicable as it is a subject formed by different ideas merged and put into complex formulas. By applying these principles, mathematicians are discovering new facts through rethinking about known information. Ben...
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
Throughout math, there are many patterns of numbers that have special and distinct properties. There are even numbers, primes, odd numbers, multiples of four, eight, seven, ten, etc. One important and strange pattern of numbers is the set of Fibonacci numbers. This is the sequence of numbers that follow in this pattern: 1, 1, 2, 3, 5, 8, 13, 21, etc. The idea is that each number is the sum of its previous two numbers (n=[n-1]+[n-2]) (Kreith). The Fibonacci numbers appear in various topics of math, such as Pascal?s Triangle and the Golden Ratio/Section. It falls under number theory, which is the study of whole or rational numbers. Number Theory develops theories, simple equations, and uses special tools to find specific numbers. Some topic examples from number theory are the Euclidean Algorithm, Fermat?s Little Theorem, and Prime Numbers.