Science Focus (Issue 23)

11 唸出數列之必要:聽射性數列 Needs to Be Said elements. For example, uranium (3) decays into protactinium (13), then into thorium (1113), and so on. One might expect a lighter element to have a longer string under audioactive decay. However, sometimes an element also decays into a combination of “shorter” elements, which is why some elements are shorter than their predecessors [3] – for instance, promethium (132) fol lows samarium (311332). The next number after 311332 should be 13212312, but it can split into three lighter elements, promethium (132), calcium (12) and zinc (312). In addition, the original starting point of 1, 11, 21, 1211 until 13112221 are referred to as “primordial elements” because they can’t be split either but are not involved in every possible sequence [5]. Conway focused on these 92 numbers, again, because they are “sufficiently general” mathematically; this means they can tell us something interesting about the sequence no matter which starting point we have. It should be possible for you to guess that since al l elements occur in a decay process, any possible decay process wi l l eventually result in only these 92 elements. But as al l mathemat icians know, a guess is not enough. Conway proved this result over about a month of work with assistance from a fel low mathematician and cal led it the “Cosmological Theorem” [3]. Soon later another simpler proof was announced, but unfortunately both proofs were not published. Eventual ly the result was proven again by several others [6]. A consequence of this theorem is that the number of digits of successive numbers increases at a constant rate for all sequences