On the Waring-Goldbach problem for fourth and fifth powers

It is shown that every sufficiently large integer congruent to 14 modulo 240 may be written as the sum of 14 fourth powers of prime numbers, and that every sufficiently large odd integer may be written as the sum of 21 fifth powers of prime numbers. The respective implicit bounds 14 and 21 improve on the previous bounds 15 (following from work of Davenport) and 23 (due to Thanigasalam). These conclusions are established through the medium of the Hardy-Littlewood method, the proofs being somewhat novel in their use of estimates stemming directly from exponential sums over prime numbers in combination with the linear sieve, rather than the conventional methods which ‘waste’ a variable or two by throwing minor arc estimates down to an auxiliary mean value estimate based on variables not restricted to be prime numbers. In the work on fifth powers, a switching principle is applied to a cognate problem involving almost primes in order to obtain the desired conclusion involving prime numbers alone.