Waring's problem

In number theory, Waring's problem is a fascinating question about how numbers can be broken down into sums of powers. It asks whether every natural number can be written as the sum of a certain number of other numbers raised to a specific power. For example, we know that any whole number can be expressed as the sum of at most four squares, nine cubes, or nineteen numbers raised to the fourth power.

This problem was first suggested in 1770 by a mathematician named Edward Waring, and it is named after him. Many years later, in 1909, another mathematician named Hilbert proved that the answer to Waring's question is yes. Because of its importance, Waring's problem even has its own special category in mathematics called "Waring's problem and variants."

Relationship with Lagrange's four-square theorem

Long ago, a mathematician named Diophantus wondered if every positive whole number could be written as the sum of four perfect squares. This idea was later called Bachet's conjecture, named after Claude Gaspard Bachet de Méziriac, who translated Diophantus's work in 1621. In 1770, another mathematician, Joseph-Louis Lagrange, proved this idea true in what is now called the four-square theorem.

Around the same time, in 1770, a mathematician named Edward Waring asked a bigger question. He wanted to know if every positive whole number could be written as the sum of cubes, fourth powers, and so on. He wondered if there was always a maximum number of these powered numbers needed to represent any positive whole number.

The number g(k)

For each number k, mathematicians want to know the smallest number s so that every number can be written as a sum of at most s numbers raised to the power k.

For example, every number can be written as the sum of at most four squares, nine cubes, or nineteen fourth powers. A famous result from 1770, called Lagrange's four-square theorem, tells us that four is the smallest number needed for squares. Over time, mathematicians have found ways to estimate this number s for higher powers, though some questions remain unsolved.

The number G(k)

From the work of Hardy and Littlewood, the number G(k) was studied along with g(k). G(k) is the smallest number such that every really big whole number can be written as a sum of at most s numbers raised to the power k. We know that G(1) = 1. For squares, we find that G(2) = 4. Davenport showed in 1939 that G(4) = 16. The exact value of G(k) is not known for other powers, but we do have some limits.

G(3) is at least 4. The largest number we know that cannot be written as a sum of 4 cubes is very big. The upper limit for G(3) is 7. For fourth powers, 13792 is the largest number that needs 17 of them. For fifth powers, 68578904422 is the last known number that needs 9.

Vaughan and Wooley have done much work on finding upper limits for G(k). Vinogradov improved these limits in 1947 and 1959. Karatsuba also made improvements in 1985. Wooley added more improvements, and Vaughan and Wooley's 2002 article summarized much of what was known at the time.