HMMT 二月 2003 · 冲刺赛 · 第 3 题

英文解析

3. If a and b are positive integers that can each be written as a sum of two squares, thenab is also a sum of two squares. Find the smallest positive integer c such that c = ab ,
   3 3 3 3 3 3
   where a = x + y and b = x + y each have solutions in integers ( x, y ), but c = x + ydoes not.
   Solution: 4
   3 3 3 3
   We can’t have c = 1 = 1 + 0 or c = 2 = 1 + 1 , and if c = 3, then a or b = ± 3 whichis not a sum of two cubes (otherwise, flipping signs of x and y if necessary, we wouldget either a sum of two nonnegative cubes to equal 3, which clearly does not happen,
   or a difference of two nonnegative cubes to equal 3, but the smallest difference between
   3 3
   two successive cubes ≥ 1 is 2 − 1 = 7). However, c = 4 does meet the conditions,
   3 3
   with a = b = 2 = 1 + 1 (an argument similar to the above shows that there are no
   3 3
   x, y with 4 = x + y ), so 4 is the answer.
   1 2 3