Friedman number

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 (+, -, ×, ÷) and sometimes exponentiation. For example, 347 is a Friedman number since 347 = 7³ + 4. Some 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, 3375 (sequence A036057 in OEIS)

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 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. All the expressions for nice Friedman numbers less than 10000 involve addition and subtraction. The first few 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 (A080035)

Currently, 81 zeroless pandigital Friedman numbers are known. The two most elegant 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 the 81 known zeroless pandigtal Friedman numbers is nice: 268435179 = -268 + 4(^{3*5 - 1**7}) - 9.

From the observation that all numbers of the form 25*10²ⁿ can be written as 500...0², 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.

It seems that all powers of 5 are Friedman numbers.

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.

Algorithms for finding Friedman numbers

There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number as mb + n, where b is the base and m, n are integers between -1 and b, we need only check each possible combination of m and n against the equalities mb + n == mn, mb + n == mⁿ, and mb + n == n^m to see which ones return true. We need not concern ourselves with m + n, since a little reflection will show that mb + n == m +n always returns false. From there it becomes obvious we need not concern ourselves with expressions like m - n and m/n.

Friedman numbers using Roman numerals

In a trivial sense, all Roman numerals with more than one symbol are Friedman numbers. The expression is created by simply inserting + signs into the numeral, and occasionally the - sign with slight rearrangement of the order of the symbols.

But Erich Friedman and Robert Happelberg have done some research into Roman numeral Friedman numbers for which the expression uses some of the other operators. Their first discovery was the nice Friedman number 8, since VIII = (V - I) * II. They have also found many Roman numeral Friedman numbers for which the expression uses exponentiation, e.g., 256 since CCLVI = IV^CC/L.

The difficulty of finding nontrivial Friedman numbers in Roman numerals increases not with the size of the number (as is the case with positional notation numbering systems) but with the numbers of symbols it has. So, for example, it is much tougher to figure out whether 137 (CXLVII) is a Friedman number in Roman numerals than it is to make the same determination for 1001 (MI). With Roman numerals, one can at least derive quite a few Friedman expressions from any new expression one discovers. Friedman and Happelberg have shown that any number ending in VIII is a Friedman number based on the expression given above, for instance.

Links

* Friedman Numbers, at Dr. Erich Friedman's home page