Consider the integer 28547. In the equation $28547 = (8+5)^{4}-(7 \times 2)$ expressed in base 10, both sides use the same digits.

An integer is a Friedman number if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic arithmetic operators (addition, subtraction, multiplication, division, exponentiation) interspersed. Brackets, as usual, are essential to clarify the order of operations. These numbers are named after Erich Friedman, Assoc. Professor of Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals.

When both sides use the digits in the same order, the number is called a "nice" or "strong" Friedman number. For example, $3125=(3 + [1 \times 2])^5$ .

These concepts can be transplanted into any standard positional numbering system using 0. Transplanting into the realm of Roman numerals, however, requires the addition of the extra constraint that the right hand side of the equation use at least one other operator besides addition or subtraction.

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Problem of the Month, August 2000

Sloane's A036057

Sloane's A080035