Consider a perfect square N having 1 as the first digit (reading from left).
Determine the minimum value of N such that it remains a perfect square when 1 is replaced by 2.
Find, if possible, the next higher value of N less than 1010 with this property.
Otherwise, prove its non-existence.
Note: Computer-program based methodology apart from semi-analytic solution is permissible.
for i=1:4000000
sq=char(string(i*i));
if sq(1)=='1'
sq(1)='2';
sq=str2num(sq);
sr=round(sqrt(sq));
if sr*sr==sq
disp([i*i i sq sr])
end
end
end
finds
perfect square sq.root perfect square sq.root
11025 105 21025 145
1102500 1050 2102500 1450
11390625 3375 21390625 4625
110250000 10500 210250000 14500
196700625 14025 296700625 17225
1139062500 33750 2139062500 46250
--------------------------------------------------------------
11025000000 105000 21025000000 145000
19670062500 140250 29670062500 172250
106683890625 326625 206683890625 454625
113906250000 337500 213906250000 462500
1102500000000 1050000 2102500000000 1450000
1967006250000 1402500 2967006250000 1722500
10668389062500 3266250 20668389062500 4546250
11390625000000 3375000 21390625000000 4625000
>>
Below the manually placed line lie results beyond 10^10.
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Posted by Charlie
on 2022-04-11 08:17:18 |
going farther
