It is conjectured by Professor Zhi-Wei Sun that for each given odd prime p＞100,there always exists an solution(x,y,z)∈[1,p]3 to the Pythagoras equation x2+y2=z2 such that x,y,z are quadratic residues or non-residues modulo p respec-tively(eight cases in total).In this paper,we are able to prove the above assertion for all sufficiently large primes p,and the method is based on the recent Burgess bound for character sums of forms in many variables due to Lillian B.Pierce and Junyan Xu.
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