The Ideal Waring Theorem For Exponents 401 - 200 , 000

the so-called ideal Waring theorem states that g(k) = I(k) for every integer fc è 1. The known facts are that g(k) = I(k) for k ^ 4, ^ 5 and 1 ^ k g 400. The calculations reported here extend this result up to k = 200,000. The conclusions are based on the work of Dickson [2] and Pillai [6] who proved independently for fc > 6 and k > 1, respectively, that g(k) — I(k) provided 2* ¡z q + r + 3, and [5] it has been established since that the ideal Waring theorem holds if 2* è q + r, k 9± 4, 5¿ 5. Dickson proved in addition that if 2* < q + r, k ^ 7 and / = [(f)*]