Mathematics Essay Example | Topics and Well Written Essays - 1500 words

Mathematics - Essay ExampleIf we let s stand for the number of kth powers, then g(k) is the least much(prenominal) 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 conundrum by establishing bounds, or the maximum number of powers. For instance, Liouville found that g(4) is at close to 53. The work of Hardy and Littlewood withal 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 much 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 shed been ma ny 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, 2147771272Hilberts proofread 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 hardly on k, such that each integer N A can be written in the formMany generalizations of Warings Problem have been made. For instance, there is the prime Warings problem, and generalizations of the problem to algebraic number fields and arbitrary fields. The problem known as the easier Warings 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 Warings Problem and variants.In 1742, Goldbach suggested that every