On warning's problem : g (4) < 21

Abstract

We prove that every natural number is a sum of at most 21 fourth powers of integers. This improves the corresponding result of H. E. Thomas, Jr. where 21 is replaced by 22. Using Vinogradov's version of the Hardy-Littlewood circle method, a careful estimation of the singular series and of the trigonometric sums by means of ideas of Ch'en-Ching Jun leads to the above result