翻訳と辞書 Words near each other ・ Sum of the Parts ・ Sum of two squares theorem ・ Sum Practysis of Medecyne and other Short Works ・ Sum rule ・ Sum rule in differentiation ・ Sum rule in integration ・ Sum rule in quantum mechanics ・ Sum to Infinity ・ Sum to Infinity (book) ・ Sum Ying Fung ・ Sum-free sequence ・ Sum-free set ・ Sum-frequency generation ・ Sum-of-squares optimization ・ Sum-of-the-parts analysis ・ Sum-product number ・ Sum/One ・ Suma ・ Suma (co-operative) ・ Suma (moth) ・ Suma Bhattacharya ・ Suma Chakrabarti ・ Suma Gestión Tributaria ・ Suma Josson ・ Suma Kanakala ・ Suma Kumanovo ・ Suma Municipality ・ Suma Park Dam ・ Suma people ・ Suma Rikyu Park Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sum-product number ： ウィキペディア英語版 Sum-product number A sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it algebraically, given an integer ''n'' that is ''l'' digits long in base ''b'' (with ''d''''x'' representing the ''x''th digit), if $n\; =\; (\backslash sum\_^l\; d\_i)(\backslash prod\_^l\; d\_j)$ then ''n'' is a sum-product number in base ''b''. In base 10, the only sum-product numbers are 0, 1, 135, 144 . Thus, for example, 144 is a sum-product number because 1 + 4 + 4 = 9, and 1 × 4 × 4 = 16, and 9 × 16 = 144. 1 is a sum-product number in any base, because of the multiplicative identity. 0 is also a sum-product number in any base, but no other integer with significant zeroes in the given base can be a sum-product number. 0 and 1 are also unique in being the only single-digit sum-product numbers in any given base; for any other single-digit number, the sum of the digits times the product of the digits works out to the number itself squared. Any integer shown to be a sum-product number in a given base must, by definition, also be a Harshad number in that base. In binary, 0 and 1 are the only sum-product numbers. The following table lists some sum-product numbers in a few selected bases: The finiteness of the list for base 10 was proven by David Wilson. First he proved that a base 10 sum-product number will not have more than 84 digits. Next, he ruled out numbers with significant zeroes. Thereafter he concentrated on digit products of the forms $2^i3^j7^k$ or $3^i5^j7^k$, which the previous constraints reduce to a set small enough to be testable by brute force in a reasonable period of time. From Wilson's proof, Raymond Puzio developed the proof that in any positional base system there is only a finite set of sum-product numbers. First he observed that any number ''n'' of length ''l'' must satisfy $n\; \backslash ge\; b^$. Second, since the largest digit in the base represents ''b'' - 1, the maximum possible value of the sum of digits of ''n'' is $l(b\; -\; 1)$ and the maximum possible value of the product of digits is $(b\; -\; 1)^l$. Multiplying the maximum possible sum by the maximum possible product gives $l(b\; -\; 1)^$, which is an upper bound of the value of any sum-product number of length ''l''. This suggests that $l(b\; -\; 1)^\; \backslash ge\; n\; \backslash ge\; b^$, or dividing both sides, $l(b\; -\; 1)^2\; \backslash ge\; (b\; /\; (b\; -\; 1))^$. Puzio then deduced that, because of the growth of exponential function, this inequality can only be true for values of ''l'' less than some limit, and thus that there can only be finitely many sum-product numbers ''n''. In Roman numerals, the only sum-product numbers are 1, 2, 3, and possibly 4 (if written IIII). ==External links== * PlanetMath, (proof that number of sum-product numbers in any base is finite ) 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「Sum-product number」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.