|
A Friedman number is an integer, which in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, and exponentiation. For example, 347 is a Friedman number, since 347 = 73 + 4. The first few base 10 Friedman numbers are: :25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3159 . Friedman numbers are named after Erich Friedman, an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida. ==Results== Parentheses can be used in the expressions, but only to override the default operator precedence, for example, in 1024 = (4 − 2)10. Allowing parentheses without operators would result in trivial Friedman numbers such as 24 = (24). Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 001729 = 1700 + 29. A ''nice'' or "''orderly''" Friedman number is a Friedman number where the digits in the expression can be arranged to be in the same order as in the number itself. For example, we can arrange 127 = 27 − 1 as 127 = −1 + 27. The first nice Friedman numbers are: :127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739 . Friedman's website shows around 100 zeroless pandigital Friedman numbers . Two of them are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34, both discovered by Mike Reid and Philippe Fondanaiche. Only one of them is nice: 268435179 = −268 + 4(3×5 − 17) − 9. Michael Brand proved that the density of Friedman numbers among the naturals is 1,〔Michael Brand, "Friedman numbers have density 1", ''Discrete Applied Mathematics'', 161(16–17), Nov. 2013, pp. 2389-2395.〕 which is to say that the probability of a number chosen randomly and uniformly between 1 and ''n'' to be a Friedman number tends to 1 as ''n'' tends to infinity. This result extends to Friedman numbers under any base of representation. He also proved that the same is true also for binary, ternary and quaternary orderly Friedman numbers.〔Michael Brand, "On the Density of Nice Friedmans", Oct 2013, http://arxiv.org/abs/1310.2390.〕 The case of base-10 orderly Friedman numbers is still open. From the observation that all numbers of the form 25×102''n'' can be written as 500...02 with ''n'' 0's, we can find strings of consecutive Friedman numbers. Friedman gives the example of 250068 = 5002 + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099. Fondanaiche thinks the smallest repdigit nice Friedman number is 99999999 = (9 + 9/9)9−9/9 − 9/9. Brandon Owens proved that repdigits of more than 24 digits are nice Friedman numbers in any base. Vampire numbers are a type of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 × 60. 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「Friedman number」の詳細全文を読む スポンサード リンク
|