# Definition:Friedman Number

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## Definition

A Friedman number (base $n$) is a (positive) integer which is the result of an expression in base $n$ arithmetic which contains exactly its digits.

The expression is subject to the following constraints:

$(1): \quad$ The arithmetic operators $+$, $-$, $\times$, $\div$ and exponentiation are the only operators which are allowed.
$(2): \quad$ Parentheses are allowed, but only in order to override the default operator precedence, otherwise every number would trivially be Friedman by $n = (n)$.
$(3): \quad$ Leading zeroes are not allowed, otherwise other numbers would trivially be Friedman by, for example, $011 = 10 + 1$.

## Examples

The first few base $10$ Friedman numbers are as follows:

 $\displaystyle 25$ $=$ $\displaystyle 5^2$ $\displaystyle 121$ $=$ $\displaystyle 11^2$ $\displaystyle 125$ $=$ $\displaystyle 5^{\paren {1 + 2} }$ $\displaystyle 126$ $=$ $\displaystyle 21 \times 6$ $\displaystyle 127$ $=$ $\displaystyle 2^7 − 1$ $\displaystyle 128$ $=$ $\displaystyle 2^{\paren {8 - 1} }$ $\displaystyle 153$ $=$ $\displaystyle 51 \times 3$

## Source of Name

This entry was named for Erich Friedman.