Friedman number
A Friedman number is an integer, which in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, and exponentiation. For example, 347 is a Friedman number, since 347 = 7^{3} + 4. The base 10 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, as of 2013^{[update]} an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida.
Contents
Results[edit]
Parentheses can be used in the expressions, but only to override the default operator precedence, for example, in 1024 = (4 − 2)^{10}. Allowing parentheses without operators would result in trivial Friedman numbers such as 24 = (24). Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 001729 = 1700 + 29.
The expressions of the first few Friedman number are:
number | expression | number | expression | number | expression | number | expression |
25 | 5^{2} | 127 | 2^{7}−1 | 289 | (8+9)^{2} | 688 | 8×86 |
121 | 11^{2} | 128 | 2^{8−1} | 343 | (3+4)^{3} | 736 | 3^{6}+7 |
125 | 5^{1+2} | 153 | 3×51 | 347 | 7^{3}+4 | 1022 | 2^{10}−2 |
126 | 6×21 | 216 | 6^{2+1} | 625 | 5^{6−2} | 1024 | (4−2)^{10} |
A nice or orderly 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 = 2^{7} − 1 as 127 = −1 + 2^{7}. 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 August 2013^{[update]}. Two of them are: 123456789 = ((86 + 2 × 7)^{5} − 91) / 3^{4}, and 987654321 = (8 × (97 + 6/2)^{5} + 1) / 3^{4}. 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.
From the observation that all numbers of the form 25×10^{2n} can be written as 5000...000^{2} with n 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. Friedman gives the example of 250068 = 500^{2} + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099.
There are an infinite number of prime Friedman numbers in base 10, because the numbers n×10^{14}+19683 = n×10^{6+8}+3^{9}+0+0+ . . . are Friedman numbers for all n. The numbers of this form are an arithmetic sequence a n+b where a and b are relatively prime, and therefore, by Dirichlet's theorem on arithmetic progressions, the sequence contains an infinite number of primes. It is simple to find similar arguments to show that this holds for bases other than base 10.
The smallest repdigit (and therefore nice) Friedman number in base 10 is thought to be 99999999 = (9 + 9/9)^{9−9/9} − 9/9. It has been proven that repdigits of more than 24 digits are nice Friedman numbers in any base.
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 numbers[edit]
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 = m^{n}, and mb + n = n^{m} 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 m − n and m/n.
Other bases[edit]
In base 12, the Friedman numbers less than 1000 are:
number | expression |
121 | 11^{2} |
127 | 7×21 |
135 | 5×31 |
144 | 4×41 |
163 | 3×61 |
368 | 8^{6−3} |
376 | 6×73 |
441 | (4+1)^{4} |
445 | 5^{4}+4 |
Using Roman numerals[edit]
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[edit]
- ^ Michael Brand, "Friedman numbers have density 1", Discrete Applied Mathematics, 161(16–17), Nov. 2013, pp. 2389-2395.
- ^ Michael Brand, "On the Density of Nice Friedmans", Oct 2013, https://arxiv.org/abs/1310.2390.