MathDB

2

Part of 1999 China Team Selection Test

Problems(2)

p = kq + r and a^2 | r

Source: China TST 1999, problem 2

5/22/2005
Find all prime numbers pp which satisfy the following condition: For any prime q<pq < p, if p=kq+r,0≤r<qp = kq + r, 0 \leq r < q, there does not exist an integer q>1q > 1 such that a2∣ra^{2} \mid r.
number theoryprime numbersnumber theory unsolved
China TST 1999 sequence

Source: China TST 1999, problem 5

5/22/2005
For a fixed natural number m≥2m \geq 2, prove that a.) There exists integers x1,x2,…,x2mx_1, x_2, \ldots, x_{2m} such that xixm+i=xi+1xm+i−1+1,i=1,2,…,m(∗)x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*) b.) For any set of integers {x1,x2,…,x2m\lbrace x_1, x_2, \ldots, x_{2m} which fulfils (*), an integral sequence …,y−k,…,y−1,y0,y1,…,yk,…\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots can be constructed such that ykym+k=yk+1ym+k−1+1,k=0,±1,±2,…y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots such that yi=xi,i=1,2,…,2my_i = x_i, i = 1, 2, \ldots, 2m.
calculusintegrationalgebra unsolvedalgebra