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.