some examples of Friedman numbers in Roman numerals

some examples of Friedman numbers in Roman numerals

In exploring

Friedman numbers

written with Roman numerals, some people may find it useful to add the constraint that other operators must be used in

addition

to the addition and

subtraction

operators, to avoid

obvious

and uninteresting instances like

X

⁢

V

⁢

I

⁢

I

⁢

I

=

X

⁢

I

+

V

⁢

I

⁢

I

or

X

⁢

I

⁢

X

=

X

⁢

X

-

I

. Also, to avoid confusion with the numeral for 10, the

multiplication

dot should be used instead of the multiplication cross.

A systematic search has yet to be undertaken. With Friedman numbers in positional bases, searches usually begin with

integers

in the range

b

<

x

<

b

3

. For Roman numerals, the search for Friedman numbers could be structured according to Sloane’s A003857.

0.1

Examples of regular Friedman numbers

V

⁢

I

⁢

I

⁢

I

=

I

⁢

V

⋅

I

⁢

I

is also an example of a

vampire number

in Roman numerals.

X

⁢

X

⁢

V

⁢

I

⁢

I

=

I

⁢

X

⋅

(

X

V

+

I

)

36 is the smallest one known to be a Friedman number in two different, non-equivalent ways:

X

⁢

X

⁢

X

⁢

V

⁢

I

=

V

⁢

I

X

⁢

X

X

or

I

⁢

X

⋅

X

⁢

X

V

.

75 can also be expressed in two different, non-equivalent ways:

L

⁢

X

⁢

X

⁢

V

=

L

⋅

X

⁢

V

X

or

X

⁢

V

⋅

L

X

.

L

⁢

X

⁢

X

⁢

V

⁢

I

⁢

I

⁢

I

=

(

X

⁢

V

-

I

⁢

I

)

⋅

(

L

X

+

I

)

C

⁢

C

⁢

X

⁢

L

⁢

I

⁢

I

⁢

I

=

I

⁢

I

⁢

I

C

⁢

C

X

⁢

L

The largest known might be

M

⁢

M

⁢

C

⁢

L

⁢

X

⁢

X

⁢

X

⁢

V

⁢

I

⁢

I

=

(

C

⁢

X

⁢

X

L

⁢

X

+

M

M

)

V

⁢

I

⁢

I

.

0.2

Examples of nice Friedman numbers

The first one found was

V

⁢

I

⁢

I

⁢

I

=

(

V

-

I

)

⋅

I

⁢

I

.

Erich Friedman himself prefers to call these ”strong” Friedman numbers, but Sloane’s

Online Encyclopedia of Integer Sequences

, because of its ”nice” keyword has pretty much codified the term ”nice.”

0.3

Examples of derived Friedman numbers

Because of the nature of Roman numerals, finding a small Roman numeral Friedman number usually allows one to derive many others. To give just two examples:

X

⁢

V

⁢

I

⁢

I

⁢

I

=

X

+

(

I

⁢

V

⋅

I

⁢

I

)

,

X

⁢

X

⁢

V

⁢

I

⁢

I

⁢

I

=

X

⁢

X

+

(

I

⁢

V

⋅

I

⁢

I

)

,

…

and

C

⁢

X

⁢

X

⁢

V

⁢

I

⁢

I

=

C

+

I

⁢

X

⋅

(

X

V

+

I

)

,

C

⁢

C

⁢

X

⁢

X

⁢

V

⁢

I

⁢

I

=

C

⁢

C

+

I

⁢

X

⋅

(

X

V

+

I

)

,

…

0.4

Examples of contrived Friedman numbers

I call these ”contrived” because using 1 as either a base or an exponent has the effect of making one of the values irrelevant.

X

⁢

L

⁢

I

⁢

X

=

L

-

I

X

⁢

X

or

L

I

-

X

X

L

⁢

X

⁢

X

⁢

X

⁢

I

⁢

X

=

X

⋅

(

X

-

I

L

)

-

X

X

Title

some examples of Friedman numbers in Roman numerals

Canonical

name

SomeExamplesOfFriedmanNumbersInRomanNumerals

Date of creation

2013-03-22 15:47:12

Last modified on

2013-03-22 15:47:12

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

6

Author

Mravinci (12996)

Entry type

Example

Classification

msc 11A63

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