㈡透过启发式论证看我们面临的困境
很难预料,入门如此容易的问题,几十年后依然令数学家与众多研究者苦不堪言。
“3N+1猜想之所以难以攻克,原因就在于对一般的n∈N,n的迭代轨迹序列中的元素排列杂乱无章,无规则可循,从而使得n的完全停止次数随着n的变化情况无法掌握。”[3]
“we show that any proof of the Collatz 3n+1 Conjecture must have an infinite number of lines. Therefore, no formal proof is possible. ”[6]
“We face this dilemma: On the one hand, to the extent that the problem has structure, we can analyze it—yet it is precisely this structure that seems to prevent us from proving that it behaves “randomly”. On the other hand, to the extent that the problem is structureless and “random,” we have nothing to analyze and consequently cannot rigorously prove anything.”[2]