Representation with distinct egyptian fractions with “small” denominators
$begingroup$
Suppose, I have a rational number $r$ with $0<r<1$, for example $$r=frac{53143}{274851}$$
The goal is to write $r$ as a sum of DISTINCT egyptian fractions (fractions with numerator $1$).
The greedy method (always take the smallest possible denominator) leads to the numbers
$$[6, 38, 2706, 34382456, 4763382302120345, 385726786254606395971426367503080]$$
That means $$r=frac{1}{6}+frac{1}{38}+frac{1}{2706}+frac{1}{34382456}+frac{1}{4763382302120345}+frac{1}{385726786254606395971426367503080}$$
The greedy methods usually leads to very large denominators. I would like to have much smaller denominators. I read somewhere that no efficient algorithm is known to find the optimal representation in the sense, that the largest occuring denominator is as small as possible. But I am looking for a method finding a "good" representation in that sense.
A much better solution in the given example would be :
$$[8, 18, 80, 3792, 30539, 48251620]$$
The maximum entry has only $8$ instead of $33$ digits.
Any ideas ?
number-theory summation fractions
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
$endgroup$
add a comment |
$begingroup$
Suppose, I have a rational number $r$ with $0<r<1$, for example $$r=frac{53143}{274851}$$
The goal is to write $r$ as a sum of DISTINCT egyptian fractions (fractions with numerator $1$).
The greedy method (always take the smallest possible denominator) leads to the numbers
$$[6, 38, 2706, 34382456, 4763382302120345, 385726786254606395971426367503080]$$
That means $$r=frac{1}{6}+frac{1}{38}+frac{1}{2706}+frac{1}{34382456}+frac{1}{4763382302120345}+frac{1}{385726786254606395971426367503080}$$
The greedy methods usually leads to very large denominators. I would like to have much smaller denominators. I read somewhere that no efficient algorithm is known to find the optimal representation in the sense, that the largest occuring denominator is as small as possible. But I am looking for a method finding a "good" representation in that sense.
A much better solution in the given example would be :
$$[8, 18, 80, 3792, 30539, 48251620]$$
The maximum entry has only $8$ instead of $33$ digits.
Any ideas ?
number-theory summation fractions
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
$endgroup$
$begingroup$
$$[6, 40, 610, 30539, 335929, 366468, 483120, 488624, 549702, 610780]$$ is an even better representation with $10$ terms, but maximum entry only $6$ digits long
$endgroup$
– Peter
Dec 15 '16 at 13:50
add a comment |
$begingroup$
Suppose, I have a rational number $r$ with $0<r<1$, for example $$r=frac{53143}{274851}$$
The goal is to write $r$ as a sum of DISTINCT egyptian fractions (fractions with numerator $1$).
The greedy method (always take the smallest possible denominator) leads to the numbers
$$[6, 38, 2706, 34382456, 4763382302120345, 385726786254606395971426367503080]$$
That means $$r=frac{1}{6}+frac{1}{38}+frac{1}{2706}+frac{1}{34382456}+frac{1}{4763382302120345}+frac{1}{385726786254606395971426367503080}$$
The greedy methods usually leads to very large denominators. I would like to have much smaller denominators. I read somewhere that no efficient algorithm is known to find the optimal representation in the sense, that the largest occuring denominator is as small as possible. But I am looking for a method finding a "good" representation in that sense.
A much better solution in the given example would be :
$$[8, 18, 80, 3792, 30539, 48251620]$$
The maximum entry has only $8$ instead of $33$ digits.
Any ideas ?
number-theory summation fractions
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
$endgroup$
Suppose, I have a rational number $r$ with $0<r<1$, for example $$r=frac{53143}{274851}$$
The goal is to write $r$ as a sum of DISTINCT egyptian fractions (fractions with numerator $1$).
The greedy method (always take the smallest possible denominator) leads to the numbers
$$[6, 38, 2706, 34382456, 4763382302120345, 385726786254606395971426367503080]$$
That means $$r=frac{1}{6}+frac{1}{38}+frac{1}{2706}+frac{1}{34382456}+frac{1}{4763382302120345}+frac{1}{385726786254606395971426367503080}$$
The greedy methods usually leads to very large denominators. I would like to have much smaller denominators. I read somewhere that no efficient algorithm is known to find the optimal representation in the sense, that the largest occuring denominator is as small as possible. But I am looking for a method finding a "good" representation in that sense.
A much better solution in the given example would be :
$$[8, 18, 80, 3792, 30539, 48251620]$$
The maximum entry has only $8$ instead of $33$ digits.
Any ideas ?
number-theory summation fractions
number-theory summation fractions
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
edited Dec 13 '16 at 17:15
Peter
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
asked Dec 13 '16 at 16:59
PeterPeter
49.3k1240138
49.3k1240138
$begingroup$
$$[6, 40, 610, 30539, 335929, 366468, 483120, 488624, 549702, 610780]$$ is an even better representation with $10$ terms, but maximum entry only $6$ digits long
$endgroup$
– Peter
Dec 15 '16 at 13:50
add a comment |
$begingroup$
$$[6, 40, 610, 30539, 335929, 366468, 483120, 488624, 549702, 610780]$$ is an even better representation with $10$ terms, but maximum entry only $6$ digits long
$endgroup$
– Peter
Dec 15 '16 at 13:50
$begingroup$
$$[6, 40, 610, 30539, 335929, 366468, 483120, 488624, 549702, 610780]$$ is an even better representation with $10$ terms, but maximum entry only $6$ digits long
$endgroup$
– Peter
Dec 15 '16 at 13:50
$begingroup$
$$[6, 40, 610, 30539, 335929, 366468, 483120, 488624, 549702, 610780]$$ is an even better representation with $10$ terms, but maximum entry only $6$ digits long
$endgroup$
– Peter
Dec 15 '16 at 13:50
add a comment |
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$begingroup$
$$[6, 40, 610, 30539, 335929, 366468, 483120, 488624, 549702, 610780]$$ is an even better representation with $10$ terms, but maximum entry only $6$ digits long
$endgroup$
– Peter
Dec 15 '16 at 13:50