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A strictly non-palindromic number is an integer ''n'' that is not palindromic in any numeral system with a base ''b'' in the range 2 ≤ ''b'' ≤ ''n'' − 2. For example, the number six is written as 110 in base 2, 20 in base 3 and 12 in base 4, none of which is a palindrome—so 6 is strictly non-palindromic. For another example, the number 167 written in base ''b'' (2 ≤ ''b'' ≤ 165) is: and none of which is a palindrome, so 167 is also a strictly non-palindromic number. The sequence of strictly non-palindromic numbers starts: :0, 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, ... To test whether a number ''n'' is strictly non-palindromic, it must be verified that ''n'' is non-palindromic in all bases up to ''n'' − 2. The reasons for this upper limit are: *any ''n'' ≥ 2 is written 11 in base ''n'' − 1, so ''n'' is palindromic in base ''n'' − 1; *any ''n'' ≥ 2 is written 10 in base ''n'', so any ''n'' is non-palindromic in base ''n''; *any ''n'' ≥ 1 is a single-digit number in any base ''b'' > ''n'', so any ''n'' is palindromic in all such bases. Thus it can be seen that the upper limit of ''n'' − 2 is necessary to obtain a mathematically "interesting" definition. For example, 167 will be written as: (if ''b'' > 165) For ''n'' < 4 the range of bases is empty, so these numbers are strictly non-palindromic in a trivial way. ==Properties== All strictly non-palindromic numbers beyond 6 are prime. To see why composite ''n'' > 6 cannot be strictly non-palindromic, for each such ''n'' a base ''b'' can be shown to exist where ''n'' is palindromic. * If ''n'' is even, then ''n'' is written 22 (a palindrome) in base ''b'' = ''n''/2 − 1. Otherwise ''n'' is odd. Write ''n'' = ''p'' · ''m'', where ''p'' is the smallest prime factor of ''n''. Then clearly ''p'' ≤ ''m''. * If ''p'' = ''m'' = 3, then ''n'' = 9 is written 1001 (a palindrome) in base ''b'' = 2. * If ''p'' = ''m'' > 3, then ''n'' is written 121 (a palindrome) in base ''b'' = ''p'' − 1. Otherwise ''p'' < ''m'' − 1. The case ''p'' = ''m'' − 1 cannot occur because both ''p'' and ''m'' are odd. * Then ''n'' is written ''pp'' (the two-digit number with each digit equal to ''p'', a palindrome) in base ''b'' = ''m'' − 1. The reader can easily verify that in each case (1) the base ''b'' is in the range 2 ≤ ''b'' ≤ ''n'' − 2, and (2) the digits ''a''''i'' of each palindrome are in the range 0 ≤ ''a''''i'' < ''b'', given that ''n'' > 6. These conditions may fail if ''n'' ≤ 6, which explains why the non-prime numbers 1, 4 and 6 are strictly non-palindromic nevertheless. Therefore, all strictly non-palindromic ''n'' > 6 are prime. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Strictly non-palindromic number」の詳細全文を読む スポンサード リンク
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