模5不同余平面格点的形心问题研究

对平面格点进行模5运算，建立了由25个不同剩余类格点排成5个行环、5个列环的环面格子网，称为模5环面，记为Z5^2．讨论了Z5^2上了格点之间、行之间、列之间、对角线之间的对称性，根据这些对称性得出了Z5^2上形心仍为格点的5个格点的分布情形．证明了当格点模5不同余时，任意9个格点中，必有5个格点其形心仍为格点，即公式n（5）=9成立．

Study on the centroid of lattice points in a plane when lattice points are different after mod 5

Using modulo 5(mod 5)operation of the lattice points in the plane,25 lattice points are arranged into a ring surface lattice with 5 row rings and 5 column rings.The ring surface is called mod 5 ring surface,marked Z2_5.Symmetry between lattice points,between rows,between columns,and between diagonals are discussed,respectively.Based on these symmetries,it is pointed out that the distributing law of the 5!lattice points on Z2_5 whose centroid on is still a lattice point.It is proved that when lattice points have different remainders after mod 5 operation,in any 9 lattice points,there exist 5 lattice points whose centroid is still a lattice point,the formula n(5)=9 found.