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The $ k $-page book crossing number of the graph $ G $, denoted by $ \\nu_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ \\nu_k(K_n) $ for all $ k\\geq 14 $ and determine $ \\nu_k(K_n) $ whenever $ 2 < n/k \\leq 3 $. 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The $ k $-page book crossing number of the graph $ G $, denoted by $ \\nu_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ \\nu_k(K_n) $ for all $ k\\geq 14 $ and determine $ \\nu_k(K_n) $ whenever $ 2 < n/k \\leq 3 $. 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