Waring's problem

The Waring problem is a problem of number theory. It generalizes the four- square theorem, which states that every natural number can be represented as a sum of at most four square numbers. In his work Meditations algebraicae (1770 ) presented Edward Waring on the assumption that there must be such a maximum number for all exponents k. The Waring problem is now solved.

The Waring problem

Waring generalization states that for every natural exponent k is a natural number g exists so that any natural number can be represented as a sum of at most g k- th powers. In other words, it comes to specify the minimum number of summands, which is necessary so that every natural number can be represented as the sum of numbers with the exponent k. For example, says the four- square theorem that every natural number as a sum of at most four square numbers (ie, g = 4 and k = 2) is representable.

G (2 ) = 4, for example 7 = 2 ² 1 ² 1 ² 1 ² evidence

Waring's conjecture that a number g exists for every k, ie, a maximum number of summands is findable for all exponents was proved in 1909 by David Hilbert. The message is therefore sometimes referred to as a set of Waring Hilbert. The Hilbert's proof was simplified in 1912 by Robert Remak and Erik Stridsberg. An elementary proof of the Waring problem, who used other ideas as Hilbert, Yuri Vladimirovich Linnik delivered in 1942 using results of Lew Schnirelman.

The smallest possible number g for an exponent k is referred to as g (k). It is g (1 ) = 1 calculations show that the number 7 requires four squares, 23 needed 9 cubes, and 79 needed 19 fourth powers. Waring assumed that these values ​​are the highest, that is g (2 ) = 4, g ( 3) = 9, and g ( 4) = 19

Thanks to the four - square theorem is proved g (2 ) = 4. The fact that g (3 ) = 9, was proved in 1909 to 1912 by Arthur Aubrey J. Kempner and Wieferich ( 1880-1973 ). Edmund Landau was also in 1909 show that only finitely many natural numbers need nine cubes, each sufficiently large number that is the sum of eight or less cubes can be represented, and Leonard E. Dickson found in 1939 that 23 and 239 are the only two numbers, which actually require nine cubes. Arthur Wieferich already guessed that in fact only 15 numbers eight and only 121 numbers require seven cubes. Today, it is generally believed that one needed only for numbers ≤ 454 eight cubes for numbers ≤ 8,042 and seven cubes for numbers ≤ 1.29074 million six cubes, so all sufficiently large numbers are represented as the sum of five cubes. The proof of the seven- cubes - set could cause the first 1941 Yuri Linnik, by George Leo Watson 1951, he was greatly simplified. g ( 4) = 19 was shown in 1986 by Ramachandran Balasubramanian, François Dress and Jean -Marc Deshouillers. Already since 1939 we also know that every sufficiently large number is represented as the sum of sixteen Biquadraten, the set of numbers that actually need 17, 18 or 19 four potencies, ie finite. This value can not be improved. g ( 5) = 37 was demonstrated in 1964 by Chen Jingrun. S. Pillai Sivasankaranarayana 1940 showed that g ( 6) = 73 And Leonard E. Dickson 1937, that g ( 7) = 143

Solution formula

Through the work of Dickson, Pillai, RK Rubugunday and Ivan M. Niven are now all other g (k ) is also known. Their formula includes two cases being assumed that the second case, for any k occurs. In the first case, the formula is:

G (k ) 1 4 9 19 37 73 143 279 548 1079 ... k 1 2 3 4 5 6 7 8 9 10 ... ( Sequence A002804 in OEIS )

For larger K the number can be estimated by g (k) ≈ 2K.

A minimum number (k)

Each of a minimum number required for the maximum number of g of summands

A (k ) 1 7 23 79 223 703 2175 6399 19455 58367 ... g (k ) 1 4 9 19 37 73 143 279 548 1079 ... k 1 2 3 4 5 6 7 8 9 10 ... ( Sequence A018886 in OEIS )

Example for k = 3: So each number is represented as the sum of at most 9 three powers ( cubes ). 23 is the first number to its representation requires nine cubes (2 ³ 2 ³ 1 ³ 1 ³ 1 ³ 1 ³ 1 ³ 1 ³ 1 ³).

Sources and Literature

• Edward Waring: Meditations algebraicae. Cambridge 31782nd
• Dennis Weeks ( ed.): Meditations algebraicae. An English translation of the work of Edward Waring. Providence: . American Mathematical Society, 1991 ( ISBN 0821801694 )