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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Perfect digit-to-digit invariant ： ウィキペディア英語版 Perfect digit-to-digit invariant A perfect digit-to-digit invariant (PDDI) (also known as a Canouchi number) is a natural number that is equal to the sum of its digits each raised to a power equal to the digit. :$n\; =\; d\_k^\; +\; d\_^\; +\; d\_1^\backslash ,.$ 0 and 1 are PDDIs in any base (using the convention that 00 = 0). Apart from 0 and 1 there are only two other PDDIs in the decimal system, 3435 and 438579088 . Note that the second of these is only a PDDI under the convention that 00 = 0, but this is standard usage in this area.〔(Narcisstic Number ), Harvey Heinz〕 :$3^3\; +\; 4^4\; +\; 3^3\; +\; 5^5\; =\; 27\; +\; 256\; +\; 27\; +\; 3125\; =\; 3435$ : :$4^4\; +\; 3^3\; +\; 8^8\; +\; 5^5\; +\; 7^7\; +\; 9^9\; +\; 0^0\; +\; 8^8\; +\; 8^8$ ::$=\; 256\; +\; 27\; +\; 16777216\; +\; 3125\; +\; 823543\; +\; 387420489\; +\; 0\; +\; 16777216\; +\; 16777216\; =\; 438579088$ More generally, there are finitely many PDDIs in any base. This can be proved as follows: :Let $b$ be a base. Every PDDI $n$ in base $b$ is equal to the sum of its digits each raised to a power equal to the digit. This sum is less than or equal to $a(b-1)^$, where $a$ is the number of digits in $n$, because $b-1$ is the largest possible digit in base $b$. Thus, ::$a(b-1)^\backslash geq\; n\backslash geq\; b^.$ :The expression $a(b-1)^$ increases linearly with respect to $a$, whereas the expression $b^$ increases exponentially with respect to $a$. So there is some $k>0$ such that :: : There are finitely many natural numbers $n$ with fewer than k digits, so there are finitely many natural numbers $n$ satisfying the first inequality. Thus, there are only finitely many PDDIs in base $b$. In base 2 the only PDDI is 1. In base 3 there are 3 PDDIs, namely 1, 12 and 22. (1, 5, 8 in decimals) In base 4 there are also 3 PDDIs, namely 1, 131 and 313. (1, 29, 55 in decimals) In base 5 there are none except for the trivial case 1. In base 6 there are 3 PDDIs, namely 1, 22352 and 23452. (1, 3164, 3416 in decimals) In base 7 there are 2 PDDIs, namely 1 and 13454. (1, 3665 in decimals) In base 8 there is again only the trivial case 1. In base 9 there are 4 PDDIs, namely 1, 31, 156262 and 1656547. (1, 28, 96446, 923362 in decimals) ==References== 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「Perfect digit-to-digit invariant」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.