Hua’s theorem for five almost equal prime squares

Abstract.

Let p _i , 1≦ i ≦ 5, be prime numbers. It is proved that every sufficiently large integer N that satisfies N ≡ 5(mod 24) can be written as N  =  p²₁ + p²₂ + p²₃ + p²₄ + p²₅, where $|\sqrt {\frac{N} {5}} - p_i | \leqq N^{\frac{1} {2} - \frac{9} {{280}} + \varepsilon } .$