Download file to see previous pages...
If we let "s" stand for the number of kth powers, then g(k) is the least such "s" powers. Some examples of g(k) are: g(1) = 1; g(2) = 4, since from Lagranges 4-square theorem, every natural number is the sum of atleast 4 squares. In addition it was found that 7 requires 4 squares and 23 requires 9 cubes.
Progress was made on Warings Problem by establishing bounds, or the maximum number of powers. For instance, Liouville found that g(4) is at most 53. The work of Hardy and Littlewood also led to other bounds; in particular, they found the upper bound for g(k) to be O(k2k+1).
The work of Hardy and Littlewood also led to the realization that the number G(k) is more fundamental than g(k). Here, G(k) is the least positive integer s such that every sufficiently large integer (greater than some constant) is a sum of at most s kth powers of positive integers. A formula for the exact value of G(k) for all k has not been found, but there have been many bounds established.
1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, 16794048, 33579681, 67146738, 134274541, 268520676, 536998744, 1073933573, 2147771272
Hilberts proof of Warings Problem for all positive k can be seen as proving an equivalent theorem: There are positive integers A and M and positive rationals 1, ..., M, depending only on k, such that each integer N A can be written in the form
Many generalizations of Waring’s Problem have been made. For instance, there is the prime Waring’s problem, and generalizations of the problem to algebraic number fields and arbitrary fields. The problem known as the “easier” Waring’s Problem takes the integer n to be a sequence of numbers x, each to the kth power. All of these variations have led to a Mathematics Subject Classification 11P05 entitled “Waring’s Problem and variants.”
In 1742, Goldbach suggested that every
...Download file to see next pagesRead More
Q1. (b) Solution of the set of equations using Inverse Matrix Method The augmented matrix for the given set of simultaneous equations is the following Applying the following transformations R2 R2 – R1; R3 R3 – R1 and R4 R4 – R1 Applying the following transformations R4 R4 – R2 - R1 Applying the following transformations R2 (1/p)*R2; R3 (1/q)*R3 and R4 (1/pq)*R4 Hence, the solution is a = u1 b = (u2 – u1)/p c = (u3 – u1)/q and d = (u4 – u3 – u2 + u1)/pq Q1.
However, despite the existence of equality and justice provisions in Basic law, women still encounter discrimination in the recruitment process. This is because many employers fear that women will desert duties once they get children or marry (Nazir & Tomppert, 2005).
Mathematical Biology has transformed spectacularly during the past two decades when there was much development of usual models which have slackly slotted in the characteristics of the general categories of biological intricacies (Hoppensteadt and Peskin, 1993).
Mathematics not only forms the very basis of Science but is the essence for the development we have seen through the ages. The various fields of Mathematics can be categorized as quantity, space, change, structure and foundations and philosophy.
To do mathematics is to explore concepts and formulate new conjectures and establish there validity by rigorous deduction from agreed upon axioms and definition.
But if you read, you would end up earning immense wealth of information.
The way the mathematicians worked to solve problems are clearly given and in particular, it says about how mathematics has become an integral part of our day to day life. It is the perfect guide for anyone who knows well about the basics of mathematics.
Teachers should be able to gauge themselves on effectiveness and engage in activities that aim at improving their work. This could be by involving their colleagues in discussions and talks on the areas that are of concern and use of observation on colleagues with good performances.
If students in mathematics classes are to learn mathematics with understanding--a goal that is accepted by almost everyone in the current debate over the role of computational skills in mathematics classrooms--then it is important to examine examples of teaching for understanding and to analyze the roles of the teacher and the knowledge that underlies the teacher's enactments of those roles.' (Liping, 1999)
Galileo Galilei a great astronomer and mathematician once said Mathematics is the language with which God has written the universe. Mathematics in essence provides the basic learning code for all subjects of studies whether that
I studied my high school education in China and joined U.S 4 years ago as an international student. I came to the U.S. four years ago as an international student. On completion, I got an associate degree from Seattle Central Community College. In this college, I took