摘要
Let k be a positive integer. For any positive integer x =∑i=0^∞xi2^i, where xi = 0, 1,we define the weight w（x） of x by w（x） ：= ∑i=0^∞xi. For any integer t with 0 〈 t 〈 2^k- 1, let St ：= {（a,b）∈ Z^2｜a＋b≡t（mod 2^k-1）,w（a）＋w（b）〈k,0≤a,b≤2^k-2}.This paper gives explicit formulas for cardinality of St in the cases of w（t） ≤ 3 and an upper bound for cardinality of St when w（t） = 4. From this one then concludes that a conjecture proposed by Tu and Deng in 2011 is true if w（t） ≤ 4.
基金
supported partially by the National Science Foundation of China under Grant No.11371260
the Youth Foundation of Sichuan University Jinjiang College under Grant No.QJ141308
