250099 in base 10. Repdigit Friedman numbers: The smallest repdigit in base 8 that is a Friedman number is 33 = 33. The smallest repdigit in base 10 that is thought to be a Friedman number is 99999999 = (9 + 9/9)9−9/9 − 9/9. It has been proven that repdigits with at least 22 digits are nice Friedman numbers. There are an infinite number of prime Friedman numbers in all bases, because for base the numbers in base 2 in base 3 in base 4 in base 5 in base 6 for base the numbers in base 7, in base 8, in base 9, in base 10, and for base are Friedman numbers for all . The numbers of this form are an arithmetic sequence , where and are relatively prime regardless of base as and are always relatively prime, and therefore, by Dirichlet's theorem on arithmetic progressions, the sequence contains an infinite number of primes. Duodecimal In base 12, the Friedman numbers less than 1000 are: Using Roman numerals 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. Other such nontrivial examples have been found. The difficulty of finding nontrivial Friedman numbers in Roman numerals increases not with the size of the number (as is the case with positional notation numbering systems) but with the numbers of symbols it has. 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). With Roman numerals, one can at least derive quite a few Friedman expressions from any new expression one discovers. Since 8 is a nice nontrivial nice Roman numeral Friedman number, it follows that any number ending in VIII is also such a Friedman number. References External links Other Friedman numbers in The On-Line Encyclopedia of Integer Sequences Base-dependent integer sequences