Number of Friedman

In Mathematical, a number of Friedman is a Integer which, in a given base, is the result of an expression using its own figures in combination with one of the four operations arithmetic basic ( $+, -, \ times, \ frac {.}{.}\,$ ) and some time the Exponentiation. For example, 347 is a number of Friedman since 347 = 7³ + 4. The first small numbers of Friedman in bases 10 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, 3125,3159

The brackets can be used in the expressions, but only to isolate an expression compared to an operator, for example, in 1024 = (4 - 2) ¹⁰. To allow the brackets without the operators would give fictitious numbers of Friedman such as 24 = (24). The zero not-significant ones are not used, since that would give fictitious numbers of Friedman, such as 001729 = 1700 + 29.

Currently, two numbers of pandigitaux Friedman without zeros are known:

123456789 = \ frac {({86 + 2 * 7} ^5 - 91)}{34}

, and:

987654321 = 8 \ times \ frac {(97 + {\ frac {6} {2}} ^5 + 1)}{3^4}

, both discovered by Mike Reid and Philippe Fondanaiche.

From the observation that all the powers of 5 appear to be numbers of Friedman, we can find chains of consecutive numbers of Friedman. Friedman gives the example of $250068 = 500^2 + 68 \,$ , from which we can easily deduce the interval from consecutive numbers of Friedman from 250010 to 250099.

A number of Friedman pleasant is a number of Friedman where the figures in the expression can be arranged in the same order as in the number itself. For example, we can arrange 127 = 2⁷ - 1 like 127 = -1 + 2⁷. All the expressions for the pleasant numbers of Friedman lower than 10000 imply the addition and the subtraction. The first pleasant small numbers of Frieman 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

Fondanaiche thinks that smallest uniform Nombre of pleasant Friedman is 99999999 = (9 + 9/9) ^9-9/9 - 9/9. Brandon Owens showed that the uniform numbers of more than 24 digits are numbers of pleasant Friedman in any base.

Algorithms to find numbers of Friedman

There exist generally less numbers of Friedman with two digits than with three digits and more in any base given, but those with two digits are easier to find. If we represent a number with two digits in the form mb + N , where B is the base and m , N of the integers ranging between -1 and B , we need to check only each possible conbinaison of m and N compared to the equalities:

m.b + N = m.n \,

m.b + N = m^n \,

, and

m.b + N = n^m \,

to see which remains true. We do not need to worry us m + N , since a small reflection will show us that mb + N = m + N will be always false. From there, it becomes obvious that we do not have to worry us expressions such as m - N and m / N .

When we treat the numbers with three digits, the concept remains the same one, only it exists more expressions possible to check. By representing a number with three digits in the form $k.b^2 + m.b + N \,$ , there exist more expressions to be checked, to start,

k^m + N \,

k^n + m \,

k^ {m+n} \,

n \ times (k.b + m) \,

, etc

Numbers of Friedman expressed in Roman numerals

In a commonplace direction, all the numbers expressed in Roman numerals with more than one symbol are numbers of Friedman. The expression east creates while simply inserting the signs + in the expression, and occasionally the sign - with a light rearrangement in the order of the symbols.

But Erich Friedman and Robert Happleberg made certain research on the numbers expressed in Roman numerals for which the expression uses other operators that + and -. Their first discovered was the number of Friedman pleasant 8, since VIII = (V - I) * II. They also found many numbers of Friedman expressed in Roman numerals for which the expression uses the exponentiation, for example 256 since $CCLVI = IV^ {\ frac {DC} {L}} \,$ .

The difficulty to find numbers of Friedman expressed in not-commonplace Roman numerals does not increase with the size of the number (as it is the case with the systems of numbers with positional Notation) but with the numbers of symbols which it has. Thus, for example, it is more painful to know if 137 (CXLVII) is a number of Friedman expressed in Roman numeral to make the same determination for 1001 (SEMI). With the numbers expressed in Roman numerals, one can at least make derive certain expressions from Friedman from which one can discover others. Friedman and Happleberg showed that any number finishing by VIII is a number of Friedman based on the expression given above, for example.

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