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 The integers are commonly denoted by the above symbol. ... In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

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) 25 (twenty-five) is the natural number following 24 and preceding 26. ... 121 is the natural number following 120 and preceding 122. ... 125 is the natural number following 124 and preceding 126. ... 126 is the natural number following 125 and preceding 127. ... 127 is the natural number following 126 and preceding 128. ... 128 is the natural number following 127 and preceding 129. ... One hundred fifty-three is the natural number following one hundred fifty-two and preceding one hundred fifty-four. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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 zeroes cannot be used, since that would also result in trivial Friedman numbers, such as 001729 = 1700 + 29.

Currently, two zeroless pandigital Friedman numbers 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. A pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. ...

From the observation that all powers of 5 appear to be Friedman numbers, 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 250010 to 250099.

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. 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)

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. In recreational mathematics, a repdigit is a natural number composed of repeated instances of the same digit, most often in the decimal numeral system. ...

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.

When we get to 3-digit numbers, the concept remains the same, only that there are more possible expressions to check. Representing a three digit number as kb² + mb + n, there are more expressions to check, for starters, k^m + n, kⁿ + m, k^{m + n}, n * (kb + m), etc.

In any given base b, one of the smallest (if not the smallest) Friedman numbers is the square of the number represented by the digit grouping 11, which is represented by the digit grouping 121 in bases 3 and above, having the Friedman expression 121 = 11², or to put it algebraically, (b + 1)² = b² + 2b + 1.

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. The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ...

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. Positional notation is a system in which each position has a value represented by a unique symbol or character. ...

External links

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