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. 2000 Mathematics Subject Classification:
11P05, 11N36, 11L15, 11P55.