Download POTW #14-01 Aliquot Fractions

POTW #14-01 Aliquot Fractions
Finding A “Smallest”Solution
John Snyder, FSA
August 16, 2013
17:15 EST
Problem
Every proper fraction can be expressed as the sum of a number of distinct aliquot fractions; i.e., fractions with unit (the number 1) numerators. What is the ”smallest”fraction (sum of numerator and denominator a minimum) which requires four such fractions for its expression? As an example,
1 5 + 1 6 + 1 7 + 1 8 = 533 840 . Of course, 533 + 840 = 1373, a very large number, indeed. This
addition must consist of four different, positive aliquot fractions. Can you find a ”smaller”fraction?
Solution
Summary
The best solution is:
1
1
+
2
1
+
6
1
+
22
8
=
66
11
We call this solution “best”because 8 11 is the proper fraction having the smallest sum of its numerator
and denominator which requires four aliquot fractions for its expression. In addition, the above four
aliquot fractions may be termed a best solution because the sum of their four denominators has the
smallest possible total of 96.
Analysis
By definition a proper fraction must have a numerator that is less than its denominator. We first need to
find the smallest such fraction, measured in terms of the sum of its numerator and denominator, which
requires at least four aliquot fractions (Egyptian fractions) for its representation. This is a somewhat
difficult problem in number theory which has only been solved up to smallest fraction requiring 7 or
fewer aliquot fractions for its representation.
Richard Guy in his book Unsolved Problems in Number Theory (see References below) states that
“Victor Meally ... noted that 2 3, 4 5 and 8 11 are the earliest numbers of the sequence that need 2, 3
and 4 unitary fractions to represent them.”We first want to confirm that 8 11 is the smallest proper
fraction that requires four unitary fractions to represent it. To do this we first write a function that will find
a solution in terms of the sum of three unitary fractions for a proper fraction having a numerator of n and
a denominator of d.
2
Richard Guy in his book Unsolved Problems in Number Theory (see References below) states that
“Victor Meally ... noted that 2 3, 4 5 and 8 11 are the earliest numbers of the sequence that need 2, 3
and 4 unitary fractions to represent them.”We first want to confirm that 8 11 is the smallest proper
fraction that requires four unitary fractions to represent it. To do this we first write a function that will find
a solution in terms of the sum of three unitary fractions for a proper fraction having a numerator of n and
a denominator of d.
AliquotFractions1401.nb
1
test3@8n_, d_<D := FindInstanceB
1
+
x
1
+
y
n
&& 1 < x < y < z, 8x, y, z<, IntegersF;
==
z
d
We test for all proper fractions representable as the sum of three unitary fractions having a numerator
plus denominator whose sum is 19 or less (this being 8 + 11 = 19).
three =
3  HarmonicMean ž Flatten@ParallelTable@8x, y, z< . Flatten@Map@test3, Map@Sort,
IntegerPartitions@t, 82<DDD, 1D, 8t, 2, 19<D, 1D  Union
1
1
,
:
1
,
18
3
17
2
, ,
14 9
6
1
, ,
13 2
1
,
16
3
,
13
6
,
11
1
,
1
,
1
,
1
,
1
,
1
,
2
,
1
,
2
,
1
,
2
,
1
,
2
,
3
,
1
,
,
15 14 13 12 11 10 9 17 8 15 7 13 6 11 16 5
1
4
3
2
3
4
1
5
4
3
5
2
5
3 4
5
,
,
, ,
,
, ,
,
, ,
, ,
, , ,
,
4 15 11 7 10 13 3 14 11 8 13 5 12 7 9 11
5 4
7
3 5
7
2
7
5 3 7 4 5 6 7 8
9
, ,
, , ,
, ,
, , , , , , , , ,
, 1>
9 7 12 5 8 11 3 10 7 4 9 5 6 7 8 9 10
We write second function that will find a solution in terms of the sum of four unitary fractions for a proper
fraction having a numerator of n and a denominator of d.
test4@8n_, d_<D :=
1 1 1 1
n
FindInstanceB + + + ==
&& 1 < w < x < y < z, 8w, x, y, z<, IntegersF;
w x y z
d
We test for all proper fractions representable as the sum of four unitary fractions having a numerator
plus denominator whose sum is also 19 or less.
four =
4  HarmonicMean ž Flatten@ParallelTable@8w, x, y, z< . Flatten@Map@test4, Map@Sort,
IntegerPartitions@t, 82<DDD, 1D, 8t, 2, 19<D, 1D  Union
1
1
,
:
18
2
,
9
1
,
2
1
,
17
3
,
13
6
,
11
1
,
1
,
1
,
1
,
1
,
1
,
1
,
2
,
1
,
2
,
1
,
2
,
1
,
2
,
3
,
1
,
16 15 14 13 12 11 10 9 17 8 15 7 13 6 11 16 5
1
4
3
2
3
4
1
5
4
3
5
2
5
3 4
5
6
,
,
, ,
,
, ,
,
, ,
, ,
, , ,
,
,
4 15 11 7 10 13 3 14 11 8 13 5 12 7 9 11 13
5 4
7
3 5
7
2
7
5
8
3 7 4 5 6 7 8
9
, ,
, , ,
, ,
, ,
, , , , , , , ,
, 1>
9 7 12 5 8 11 3 10 7 11 4 9 5 6 7 8 9 10
3
,
,
14
Taking the complement of these two lists we confirm that 8 11 is smallest proper fraction requiring four
unitary fractions in its representation.
Complement@four, threeD
8
:
>
11
Now that we know that the 8 11 is the smallest fraction, we can take the problem one step more and
find the values of the denominators of the four unitary fractions such that the sum of these denominators is minimized. A little testing shows, as we see in the next two cells, that the minimum sum of the
four denominators is 96.
AliquotFractions1401.nb
3
FindInstanceB
1
1
+
w
1
+
x
1
+
y
8
&& 1 < w < x < y < z && w + x + y + z £ 95, 8w, x, y, z<, IntegersF
==
z
11
8<
1
FindInstanceB
1
+
w
1
+
x
1
+
y
8
&& 1 < w < x < y < z && w + x + y + z £ 96,
==
z
11
8w, x, y, z<, Integers, 5F
88w ® 2, x ® 6, y ® 22, z ® 66<<
This last solution is unique. So we have found what we might call a best solution consisting of the sum
of the following four unitary fractions.
1
1
+
2 6
True
1
+
1
+
22
8
Š
66
11
References
Guy, Richard K., Unsolved Problems in Number Theory, Third Edition, Springer-Verlag, 2004, Section
D11, pp 254-255
What’
s the simplest rational not expressible as a sum of a given number of unit fractions?, MathOverflow, http://mathoverflow.net/questions/33628/whats-the-simplest-rational-not-expressible-as-a-sum-ofa-given-number-of-unit