A Friedman number is an integer, which represented in a given numeral system, is the result of a non-trivial expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, exponentiation, and concatenation.
Here, non-trivial means that at least one operation besides concatenation is used.
Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4.
For example, 347 is a Friedman number in the decimal numeral system, since 347 = 73 + 4.
The decimal Friedman numbers are: Friedman numbers are named after Erich Friedman, a now-retired mathematics professor at Stetson University and recreational mathematics enthusiast.
The first nice Friedman primes are: Michael Brand proved that the density of Friedman numbers among the naturals is 1,[1] 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 nice Friedman numbers.
[2] The case of base-10 nice Friedman numbers is still open.
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.
For example, 110012 = 25 is a Friedman number in the binary numeral system, since 11001 = 10110.
From the observation that all numbers of the form 2k × b2n can be written as k000...0002 with n 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long.
The numbers of this form are an arithmetic sequence
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.
Some research into Roman numeral Friedman numbers for which the expression uses some of the other operators has been done.
The first such nice Roman numeral Friedman number discovered was 8, since VIII = (V - I) × II.
For example, it is much tougher to figure out whether 147 (CXLVII) is a Friedman number in Roman numerals than it is to make the same determination for 1001 (MI).