Friedmann's number

In mathematics, a Friedman number is an integer which, given a base, is the result of an expression using its own digits in combination with any of the four arithmetic operations (+, -, ×, ÷) and sometimes with powers. For example, 347 is a Friedman number since 347 = 7³ + 4. The first Friedman numbers in base 10 are

25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 15035, 14035, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3... (6059 sequence A in OEIS))

Rules

Parentheses can be used in expressions, but only to alter operator precedence. For example, 1024 = (4 - 2)¹⁰. If parentheses are allowed without operators, it would result in trivial Friedman numbers, such as 24 = (24). The leading zeros are not used, since we would get, for example, 001729 = 1700 + 29, which is also a trivial Friedman number.

In fact, two pandigital Friedman numbers without zeros are known: 123456789 = ((86 + 2 * 7)⁵ - 91) / 34, and 987654321 = (8 * (97 + 6 /2)⁵ + 1) / 3⁴, both discovered by Mike Reid and Philippe Fondanaiche.

Since all powers of 5 appear to be Friedman numbers, we can find consecutive strings of Friedman numbers. Friedman himself gives the example of 250068 = 500² + 68, which can be easily deduced from Friedman's range of consecutive numbers from 250010 to 250099.

A nice Friedman number is such that the digits in the expression can be rearranged so that they are in the same order of appearance as in the number itself. For example, we can reorder 127 = 2⁷ - 1 as 127 = -1 + 2⁷. All expressions for this class of numbers less than 10000 involve addition and subtraction. The first numbers of this class 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, 19683, 19739 (sequence A080035 in OEIS))

Fondanaiche believes that the smallest of these numbers such that all their digits are the same is 99999999 = (9 + 9/9)^9-9/9 - 9/9. Brandon Owens showed that these numbers, when they have more than 24 digits, are nice Friedman numbers in any base.

Algorithms for finding Friedman Numbers

Two digits

There are usually fewer two-digit Friedman numbers than three-digits in a given base, but the former are the easiest to find. If we represent a two-digit number as mb + n, where b is the base and m and n the integers between -1 and b, we only need to check that every possible combination of m and n in the following equalities:

• mb + n ♪ mn
• mb + n ♪ mⁿ
• mb + n ♪ n^m

And see which ones are verified. We don't have to worry about m + n, since mb + n == m + n is always false. From here it is obvious that we will not be concerned with expressions such as m - n and m/n.

Three digits

The concept is the same as obtaining two-digit numbers, but now we have more expressions to check. If we represent a three-digit number as kb² + mb + n, to begin with we should check:

• k^m + n
• kⁿ + m
• k^{m + n}
• n ♪kb + m)
• etc.

In any base b, one of the smallest (if not the smallest) of the Friedman numbers is the square of the number represented by the association of digits that make up 11, represented by 121 for base 3 and higher, having the Friedman expression 121 = 11², or algebraically, 121 = 11².

Friedman numbers in Roman notation

Trivially, all Roman numerals with more than one symbol are Friedman numbers. The expression is created by inserting + signs into the numeral, and occasionally the - sign with a slight rearrangement of the symbols.
Erich Friedman and Robert Happelberg did research in this field to find expressions that used some other operator. His first discovery was the number 8, since VIII = (V - I) * II, which is also a nice Friedman Number. They found many other Friedman numbers where the expression uses exponentiation, such as 256, since CCLVI = IV^CC/L.
The difficulty in finding non-trivial Friedman numbers in Roman numerals increases not only with the size of the number (as is the case with positional notation systems), but with the number of symbols it contains. For example, it is harder to tell that 137 (CXLVII) is a Friedman number than 1001 (MI). With Roman notation, some expressions can be derived from some other that has been discovered. For example, Friedman and Happelberg showed that any number ending in VIII is a Friedman number.