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:
\(\ds 25\) | \(=\) | \(\ds 5^2\) | ||||||||||||
\(\ds 121\) | \(=\) | \(\ds 11^2\) | ||||||||||||
\(\ds 125\) | \(=\) | \(\ds 5^{\paren {1 + 2} }\) | ||||||||||||
\(\ds 126\) | \(=\) | \(\ds 21 \times 6\) | ||||||||||||
\(\ds 127\) | \(=\) | \(\ds 2^7 − 1\) | ||||||||||||
\(\ds 128\) | \(=\) | \(\ds 2^{\paren {8 - 1} }\) | ||||||||||||
\(\ds 153\) | \(=\) | \(\ds 51 \times 3\) |
Also see
- Results about Friedman numbers can be found here.
Source of Name
This entry was named for Erich Friedman.