Friedman number

A Friedman number is an integer, which represented in a given numeral system, is the result of a non-trivial expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, exponentiation, and concatenation. Here, non-trivial means that at least one operation besides concatenation is used. Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4. For example, 347 is a Friedman number in the decimal numeral system, since 347 = 73 + 4. The decimal Friedman numbers are:

25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, ... (sequence A036057 in the OEIS).

Friedman numbers are named after Erich Friedman, a now-retired mathematics professor at Stetson University, located in DeLand, Florida.

Results in base 10Edit

The expressions of the first few Friedman numbers are:

number expression number expression number expression number expression
25 52 127 27−1 289 (8+9)2 688 8×86
121 112 128 28−1 343 (3+4)3 736 36+7
125 51+2 153 3×51 347 73+4 1022 210−2
126 6×21 216 62+1 625 56−2 1024 (4−2)10

A nice Friedman number is a Friedman number where the digits in the expression can be arranged to be in the same order as in the number itself. For example, we can arrange 127 = 27 − 1 as 127 = −1 + 27. The first nice Friedman numbers are:

127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739 (sequence A080035 in the OEIS).

Friedman's website shows around 100 zeroless pandigital Friedman numbers as of April 2020. Two of them are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34. Only one of them is nice: 268435179 = −268 + 4(3×5 − 17) − 9.

Michael Brand proved that the density of Friedman numbers among the naturals is 1,[1] which is to say that the probability of a number chosen randomly and uniformly between 1 and n to be a Friedman number tends to 1 as n tends to infinity. This result extends to Friedman numbers under any base of representation. He also proved that the same is true also for binary, ternary and quaternary orderly Friedman numbers.[2] The case of base-10 orderly Friedman numbers is still open.

Vampire numbers are a subset of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 × 60.

Finding 2-digit Friedman numbersEdit

There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number as mb + n, where b is the base and m, n are integers from 0 to b−1, we need only check each possible combination of m and n against the equalities mb + n = mn, and mb + n = nm to see which ones are true. We need not concern ourselves with m + n or m × n, since these will always be smaller than mb + n when n < b. The same clearly holds for mn and m / n.

Other basesEdit

General resultsEdit

In base  ,

 

is a Friedman number (written in base   as 1mk = k × m1).[3]

In base  ,

 

is a Friedman number (written in base   as 100...00200...001 = 100..0012, with   zeroes between each nonzero number).[3]

In base  ,

 

is a Friedman number (written in base   as 2k = k2). From the observation that all numbers of the form 2k × b2n can be written as k000...0002 with n 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. For example, for  , or in base 10, 250068 = 5002 + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099 in base 10.[3]

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.[3]
  • It has been proven that repdigits with at least 22 digits are nice Friedman numbers.[3]

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.

DuodecimalEdit

In base 12, the Friedman numbers less than 1000 are:

number expression
121 112
127 7×21
135 5×31
144 4×41
163 3×61
346 34×6
368 86−3
376 6×73
441 (4+1)4
445 54+4

Using Roman numeralsEdit

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.

ReferencesEdit

  1. ^ Michael Brand, "Friedman numbers have density 1", Discrete Applied Mathematics, 161(16–17), Nov. 2013, pp. 2389-2395.
  2. ^ Michael Brand, "On the Density of Nice Friedmans", Oct 2013, https://arxiv.org/abs/1310.2390.
  3. ^ a b c d e https://erich-friedman.github.io/mathmagic/0800.html

See alsoEdit

External linksEdit