The minimum number of summands is 5. There are 37 ways if, as usual, duplicates are not allowed. If duplicates are allowed, there are only eight more ways: 45.
clearvars,clc
global depth found v egyptian level remain
syms egyptian
for depth=3:6
v=sym(21)/23;
egyptian=sym.empty; found=0; level=0; rnum=0;rden=1;
addon( )
if found>0
break
end
end
function addon( )
global depth found v egyptian level remain
if isempty(egyptian)
egyptian=sym.empty;
remain=v;
end
[numr,denr]=numden(remain);
d=floor(denr/numr);
level=level+1;
if level<=depth
while 1/sym(d)>=remain-sum(1./[sym(d+1):d+depth-level])
if remain<0
break
end
if level>1
if d<=1/egyptian(end)
d=1/egyptian(end)+1;
end
end
egyptian(end+1)=1/sym(d);
remain=remain-1/sym(d);
if remain<=0
if remain==0 && level== depth
found=found+1;
fmat=repmat(' %s',1,depth');
fprintf(['%4d ' fmat '\n'],found,egyptian);
end
else
if level<=depth
addon( )
end
end
remain=remain+1/sym(d);
egyptian(end)=[];
d=d+1;
end
end
level=level-1;
end
While the usual way of writing the egyptian fractions is to list only the denominators of the fractions, I list here the fractions themselves.
1 1/2 1/3 1/13 1/359 1/644046
2 1/2 1/3 1/13 1/360 1/107640
3 1/2 1/3 1/13 1/364 1/25116
4 1/2 1/3 1/13 1/366 1/18239
5 1/2 1/3 1/13 1/368 1/14352
6 1/2 1/3 1/13 1/390 1/4485
7 1/2 1/3 1/13 1/414 1/2691
8 1/2 1/3 1/13 1/494 1/1311
9 1/2 1/3 1/13 1/598 1/897
10 1/2 1/3 1/13 1/663 1/782
11 1/2 1/3 1/14 1/121 1/58443
12 1/2 1/3 1/14 1/123 1/6601
13 1/2 1/3 1/14 1/126 1/2898
14 1/2 1/3 1/14 1/133 1/1311
15 1/2 1/3 1/14 1/138 1/966
16 1/2 1/3 1/14 1/161 1/483
17 1/2 1/3 1/14 1/231 1/253
18 1/2 1/3 1/15 1/77 1/17710
19 1/2 1/3 1/15 1/78 1/4485
20 1/2 1/3 1/15 1/80 1/1840
21 1/2 1/3 1/15 1/85 1/782
22 1/2 1/3 1/15 1/92 1/460
23 1/2 1/3 1/15 1/110 1/253
24 1/2 1/3 1/15 1/115 1/230
25 1/2 1/3 1/16 1/60 1/1840
26 1/2 1/3 1/16 1/69 1/368
27 1/2 1/3 1/17 1/48 1/18768
28 1/2 1/3 1/17 1/51 1/782
29 1/2 1/3 1/18 1/42 1/2898
30 1/2 1/3 1/18 1/46 1/414
31 1/2 1/3 1/19 1/38 1/1311
32 1/2 1/3 1/22 1/33 1/253
33 1/2 1/3 1/23 1/28 1/1932
34 1/2 1/3 1/23 1/30 1/345
35 1/2 1/3 1/23 1/46 1/69
36 1/2 1/4 1/10 1/16 1/1840
37 1/2 1/5 1/6 1/23 1/345
If duplicate values are allowed the list expands to 45 sets:
1 1/2 1/3 1/13 1/359 1/644046
2 1/2 1/3 1/13 1/360 1/107640
3 1/2 1/3 1/13 1/364 1/25116
4 1/2 1/3 1/13 1/366 1/18239
5 1/2 1/3 1/13 1/368 1/14352
6 1/2 1/3 1/13 1/390 1/4485
7 1/2 1/3 1/13 1/414 1/2691
8 1/2 1/3 1/13 1/494 1/1311
9 1/2 1/3 1/13 1/598 1/897
10 1/2 1/3 1/13 1/663 1/782
11 1/2 1/3 1/14 1/121 1/58443
12 1/2 1/3 1/14 1/123 1/6601
13 1/2 1/3 1/14 1/126 1/2898
14 1/2 1/3 1/14 1/133 1/1311
15 1/2 1/3 1/14 1/138 1/966
16 1/2 1/3 1/14 1/161 1/483
17 1/2 1/3 1/14 1/231 1/253
18 1/2 1/3 1/15 1/77 1/17710
19 1/2 1/3 1/15 1/78 1/4485
20 1/2 1/3 1/15 1/80 1/1840
21 1/2 1/3 1/15 1/85 1/782
22 1/2 1/3 1/15 1/92 1/460
23 1/2 1/3 1/15 1/110 1/253
24 1/2 1/3 1/15 1/115 1/230
25 1/2 1/3 1/16 1/60 1/1840
26 1/2 1/3 1/16 1/69 1/368
27 1/2 1/3 1/17 1/48 1/18768
28 1/2 1/3 1/17 1/51 1/782
29 1/2 1/3 1/18 1/42 1/2898
30 1/2 1/3 1/18 1/46 1/414
31 1/2 1/3 1/19 1/38 1/1311
32 1/2 1/3 1/22 1/33 1/253
33 1/2 1/3 1/23 1/28 1/1932
34 1/2 1/3 1/23 1/30 1/345
35 1/2 1/3 1/23 1/46 1/69
36 1/2 1/4 1/10 1/16 1/1840
37 1/2 1/5 1/5 1/77 1/17710
38 1/2 1/5 1/5 1/78 1/4485
39 1/2 1/5 1/5 1/80 1/1840
40 1/2 1/5 1/5 1/85 1/782
41 1/2 1/5 1/5 1/92 1/460
42 1/2 1/5 1/5 1/110 1/253
43 1/2 1/5 1/5 1/115 1/230
44 1/2 1/5 1/6 1/23 1/345
45 1/3 1/3 1/5 1/23 1/345
(entries 37-43, and 45 have duplicated values)
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Posted by Charlie
on 2022-06-24 12:36:49 |
computer solution
