HMMT 二月 2000 · POW 赛 · 第 20 题

英文解析

20. First, we will sho w that a p ositiv e integer x is not a di eren e of squares if and only ifx 2 (mo d 4).
    2 2
    First, supp ose x = a b for some in tegers a; b , so x = ( a b )( a + b ). No w, if a b = 1,
    then a + b = 2 n + 1 for some integer n , so x = ( a b )( a + b ) = 2 n + 1, and x is o dd.
    Con v ersely , if x is o dd, then x = 2 n + 1 for some integer n , so x = 1(2 n + 1) =
    ( n + 1 n )( n + 1 + n ) = ( n + 1 ) n .22
    No w, if a b = 2, then a + b = 2 n + 2 for some integer n , so x = ( a b )( a + b ) =
    2(2 n + 2) = 4 n + 4
    Con v ersely , if x 0 (mo d 4), then x = 4 n + 4 for some integer n , so x = ( n + 2) n .22
    So, for all other ases, a b 2. If a b is ev en, then we kno w a b = 2 n for some integern , and a + b = 2 n + 2 m for some integer m , so x = ( a b )( a + b ) = 2 n (2 n + 2 m ) =
    4 nm + 4 n , so x 0 (mo d 4), and we have already o v ered this ase.2
    If a b is o dd, then we kno w a b = 2 n + 1 for some integer n , and a + b = 2 n + 2 m + 1,
    where m is some integer, so x = ( a b )( a + b ) = (2 n + 1) (2 n + 2 m + 1) = 4( n + nm +2
    n ) + 2 n + 2 m + 1 1 (mo d 4), which is a ase that we have already o v ered.
    Therefore, x is not a di eren e of t w o squares if and only if x 2 (mo d 4). So, the
    2000 th number is 4 1999 + 2 = 7998 .