Let p be a prime number. For each k, 1≤k≤p−1, there exists a unique integer denoted by k−1 such that 1≤k−1≤p−1 and k−1⋅k=1(modp). Prove that the sequence
1^{-1}, 1^{-1}+2^{-1}, 1^{-1}+2^{-1}+3^{-1}, \ldots , 1^{-1}+2^{-1}+\ldots +(p-1)^{-1}
(addition modulo p) contains at most 2p+1 distinct elements. modular arithmeticnumber theory proposednumber theory