S is a square with no repeat digits.
C is a cube with no repeat digits.
F, a pandigital fifth power, is a scramble of the concatenation of the digits of S and C.
F does have repeat digits, but there are no digits which appear in F more than twice.
None of S, C, or F can end in zero.
F is the smallest pandigital fifth power that meets these conditions.
There are several (S,C) pairs that work with this F.
Find F and list all of the (S,C) pairs (there are less than 10)
The program finds 610 valid squares and 40 valid cubes. For each pair of square and cube if creates a combined set of digits in sorted order, and sees if that ordered set of digits is listed in the list of 11 valid fifth powers where each of them has also had its digits sorted in increasing order.
clearvars,clc
squares=[]; cubes=[]; fifths={}; sortfifths={};
ansSq=vpa(1);ansCu=vpa(0);ansFi=vpa(0);
cubes=[];
for i=1:100000
is=i^2;
ds=sort(num2str(is));
if isequal(ds,unique(ds)) ...
&& mod(is,10)>0
squares(end+1)=is;
end
end
for i=1:2145
is=i^3;
ds=sort(num2str(is));
if isequal(ds,unique(ds))
cubes(end+1)=is;
end
end
for i=sym(1):sym(9997)
is=i^5;
c=char(is);
n=c-'0';
[gc,~]=groupcounts(n');
if mod(is,10)>0 && max(gc)<3 && length(setdiff( '0123456789',c))==0
fifths{end+1}=is;
sortfifths{end+1}=sort(char(is));
end
end
for s=1:length(squares)
sq=num2str(squares(s));
for c=1:length(cubes)
cu=num2str(cubes(c));
combo=sort([sq cu]);
if ismember(combo,sortfifths)
for i=1:length(sortfifths)
if isequal(combo,sortfifths{i})
fifth=fifths(i);
break
end
end
disp([squares(s) cubes(c) fifth{1}])
ansSq(end+1)=squares(s);
ansCu(end+1)=cubes(c);
ansFi(end+1)=fifth{1};
end
end
end
[ ansFi,idx]=sort(ansFi);
ansSq=ansSq(idx);
ansCu=ansCu(idx);
for i=1:length(ansFi);
fprintf('%12d %12d %22s\n',ansSq(i),ansCu(i),ansFi(i));
end
From the list below F is 21047953604832 = 462^5.
The roots (square, cube, fifth) for the requested set are:
323 302 462
352 302 462
557 302 462
1017 135 462
1197 135 462
1428 135 462
1701 135 462
1953 135 462
3018 135 462
The sorted results list is:
square cube fifth power
104329 27543608 21047953604832 \
123904 27543608 21047953604832 |
310249 27543608 21047953604832 | The F and the (S, C) pairs
1034289 2460375 21047953604832 |
1432809 2460375 21047953604832 |
2039184 2460375 21047953604832 |
2893401 2460375 21047953604832 |
3814209 2460375 21047953604832 |
9108324 2460375 21047953604832 /
198025 24137569 62854912109375
198025 32461759 62854912109375
259081 24137569 62854912109375
259081 32461759 62854912109375
819025 24137569 62854912109375
819025 32461759 62854912109375
1054729 27543608 170484759256032
1974025 27543608 170484759256032
2715904 27543608 170484759256032
4507129 27543608 170484759256032
5470921 27543608 170484759256032
7295401 27543608 170484759256032
19847025 2460375 170484759256032
57108249 2460375 170484759256032
71520849 2460375 170484759256032
80514729 2460375 170484759256032
94187025 2460375 170484759256032
102576384 592704 170484759256032
105637284 592704 170484759256032
158306724 592704 170484759256032
176305284 592704 170484759256032
180472356 592704 170484759256032
183467025 592704 170484759256032
187635204 592704 170484759256032
208571364 592704 170484759256032
218034756 592704 170484759256032
284057316 592704 170484759256032
307265841 592704 170484759256032
316057284 592704 170484759256032
430728516 592704 170484759256032
472801536 592704 170484759256032
475283601 592704 170484759256032
560837124 592704 170484759256032
570684321 592704 170484759256032
576432081 592704 170484759256032
734681025 592704 170484759256032
783104256 592704 170484759256032
825470361 592704 170484759256032
853107264 592704 170484759256032
1034289 24137569 564708431199232
1034289 32461759 564708431199232
1432809 24137569 564708431199232
1432809 32461759 564708431199232
2039184 24137569 564708431199232
2039184 32461759 564708431199232
2893401 24137569 564708431199232
2893401 32461759 564708431199232
3814209 24137569 564708431199232
3814209 32461759 564708431199232
9108324 24137569 564708431199232
9108324 32461759 564708431199232
1763584 26198073 885623410917376
7458361 26198073 885623410917376
8317456 26198073 885623410917376
102738496 2460375 2436910203746875
172843609 2460375 2436910203746875
176039824 2460375 2436910203746875
273869401 2460375 2436910203746875
420783169 2460375 2436910203746875
478603129 2460375 2436910203746875
639027841 2460375 2436910203746875
683927104 2460375 2436910203746875
740329681 2460375 2436910203746875
816930724 2460375 2436910203746875
1026753849 592704 5514047299623807
1042385796 592704 5514047299623807
1098524736 592704 5514047299623807
1237069584 592704 5514047299623807
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1278563049 592704 5514047299623807
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1382054976 592704 5514047299623807
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1503267984 592704 5514047299623807
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1547320896 592704 5514047299623807
1643897025 592704 5514047299623807
1827049536 592704 5514047299623807
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1937408256 592704 5514047299623807
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2431870596 592704 5514047299623807
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3528716409 592704 5514047299623807
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3791480625 592704 5514047299623807
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3964087521 592704 5514047299623807
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4580176329 592704 5514047299623807
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5102673489 592704 5514047299623807
5273809641 592704 5514047299623807
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5783146209 592704 5514047299623807
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6154873209 592704 5514047299623807
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8503421796 592704 5514047299623807
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9560732841 592704 5514047299623807
9614783025 592704 5514047299623807
9761835204 592704 5514047299623807
9814072356 592704 5514047299623807
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Posted by Charlie
on 2024-12-18 09:44:12 |
computer solution
