Warings Problem and Goldbachs Conjecture - Assignment Example

Waring’s Problem and Goldbach’s Conjecture 20-2008 Waring’s Problem Lagrange’s 4 square theorem s that every number can be written as the sumof four integer squares. Waring proposed a generalisation of this theorem in 1770, stating that every natural number is the sum of a fixed number g(k) of kth power integers, where k is any given positive integer and g(k) depends only on k. In this paper, we will investigate what is currently known about this problem. Warings Problem was proven for all k by Hilbert in 1909 (Ellison 1971), with the Hilbert-Waring Theorem. Prior to that, the problem was solved in the affirmative for specific ks, especially for small k. 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. Other values for the g(k)s have been found over the years: Wieferich established that g(3) = 9 between 1909-1912; as recently as 1986 it was found that g(4) = 19. Euler found a formula for each g(k): g(k) = 2k + [(3/2)k]-2 where [x] denotes the integral part of x. This conjecture holds for each positive integer k, as long as a certain condition never occurs: 2k{(3/2)k} + [(3/2)k] > 2k Using the Euler formula, the sequence of g(k) is found to be (AT&T 2007): 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 which illustrates that much progress has been made in the actual computation of g(k). 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 where ni Z+ for 1 i M. The key result of Hilbert’s proof is the following lemma: For each positive k there are positive rational numbers 0, ..., N, where N = (2k+1)(2k+4)/24, and integers 11, ..., N1, 12, ...N5 such that The set of homogeneous forms of degree 2k with 5 variables with real coefficients forms a vector space of dimension (2k+1)(2k+4)/24, which is N, the number of coefficients. Hilbert’s proof shows that we can find integers A and N0 that depend only on k, such that for integer T N0, then each integer n in the range ATk n A (T + 1)k can be written as the sum Every integer greater than A N0k is contained in such an interval, by choosing T correctly. This method then proves the theorem. 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.” The Goldbach Conjecture In 1742, Goldbach suggested that every integer greater than 2 could be written as the sum of 3 primes. However, Goldbach considered the number 1 to be prime, Euler reformulated Goldbach’s statement to propose that: all even integers greater than 2 can be written as the sum of 2 primes, and this is now known as the Goldbach Conjecture. This paper explores the progress that has been made towards the proof of this conjecture to date. The above statement is the strong/even/binary conjecture. The weak/odd/tertiary conjecture is an implication of the above, and is as follows: all odd numbers greater than 7 are the sum of three odd primes. More progress has been made in solving the weak conjecture than the strong form; the weak conjecture is considered almost solved. Expressing a Goldbach number (an integer greater or equal to four) as the sum of two primes is called the partition of the Goldbach number. For instance, the partition of 4 = 2 + 2; 6 = 3 + 3, etc. There is a heuristic argument in favor of Goldbachs conjecture that is based on the probabilistic distribution of the prime numbers. The reasoning is basically that for larger integers, there are more ways to form the partition, leading to a higher likelihood the partition will consist of prime numbers. Progress towards the solution of Goldbach’s Conjecture was first made by Brun (Kumchev & Tolev 2007). He showed that every large even number is the sum of two integers having at most 9 prime factors. Brun also found an upper bound on the number of representations of a large even integer as the sum of two primes. Brun’s results were made after he created the Brun sieve in 1915, which was based on the Legendre’s sieve of Eratosthenes, but improved on it by using inequalities, truncating a series after a number of terms to give a bound. The Brun sieve led to the theorem that there are infinitely many integers n such that n and n + 2 have at most nine prime factors; and that the sum of reciprocals of twin primes converges to Brun’s constant. The Brun sieve also began modern sieve theory. Hardy and Littlewood in the 1920s used their circle method and the Generalized Riemann Hypothesis(GRH) to prove that all but finitely many odd integers are sums of three primes and all but a certain number of even integers are sums of two primes. Schnirelmann then developed probabilistic methods in the 1930s. During this time, Vinogradov also developed his three prime theorem, which showed that every sufficiently large odd integer is the sum of three primes. It was considered significant during this time that the GRH was not used in these methods. In the 1940s Renyi provided a result that almost led to the solution of the Goldbach binary conjecture. Renyi’s result is that there is a fixed integer r such that the sequence of primes A = A(n) = {n-p: p a prime number, 2 < p < n} contains a Pr-number when n is sufficiently large. Subsequent work reduced the value of r; particularly, Chen’s result (found by Chen Jingrun) of 1973, which is as follows: There exists a constant n0, so that if n is larger or equal to n0, then where the integer n = p + P2, with p a prime number and P2 is an almost prime of order 2; r(n) is the number of representations of n in this form. Chen’s result can be written more succinctly as: Chen used sieve theory methods to obtain this result, which established that every sufficiently large even number can be written as the sum of two primes, or a prime and a semiprime (the product of two primes, also known as an almost prime). Other work similar to Chen’s includes the 1975 result of Montgomery and Vaughan that showed most even integers are the sum of two primes. They proved for positive constants c and C, and sufficiently large numbers N, there are at most CN 1-c even numbers less than N that are not the sum of two primes. There is also the 2002 result of Heath-Brown and Schlage-Puchta that every sufficiently large even integer is a sum of two primes and 13 powers of 2 (Heath-Brown & Puchta 2002). This work is an improvement on Linnik’s theorem which used K powers of 2, with K undetermined. Heath-Brown and Puchta also state the theorem that if the GRH is assumed, then only 7 powers of 2 are needed. The eventual solution of the Goldbach conjecture may involve approaches that are used on the similar Twin Prime Conjecture, proposed by Euclid around 300 BC, which states: There are infinitely many primes p such that p + 2 is also prime. In particular, both Viggo Brun and Chen Jingrun utilized the same sieve methods for treating the Twin Prime Conjecture and the Goldbach Conjecture. At the very least, advances in understanding in one of the conjectures may shed light on the other conjecture. The strong Goldbach conjecture can be proved for small n. For instance, Pipping in 1938 solved the strong conjecture for n 105. Using computers, Silva in 2007 has verified the conjecture up to n 1018 (Silva 2007). It remains a challenge for mathematicians to find a proof of the strong Goldbach conjecture, one of the oldest unsolved problems, for all n using simple methods. REFERENCES AT&T Knowledge Ventures (2007), Online Encyclopedia of Integer Sequences, entry A002804. http://www.research.att.com/~njas/sequences/A002804 Ellison, W.J.(1971, Jan). "Warings Problem." The American Mathematical Monthly, vol. 78, no 1, p10-36. Heath-Brown, D.R. and Puchta, J.C. (2002) “Integers Represented as a Sum of Primes and Powers of Two.” http://arxiv.org/PS_cache/math/pdf/0201/0201299v2.pdf Kumchev, A.M. and Tolev, D.I. (2007) "An Invitation to Additive Prime Number Theory." http://arxiv.org/PS_cache/math/pdf/0412/0412220v1.pdf Silva, Tomas Oliveira e (2007). Goldbach Conjecture Verification. http://www.ieeta.pt/~tos/goldbach.html Read More