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.
The analytic work in my previous comment very tightly restricts the search space for nontrivial solutions. So even a slow interpreter like UBASIC can find larger solutions quickly. This program quickly finds 11 nontrivial solutions up to 30 digits.
5 print=print+"output.txt"
10 for P=2 to 29
20 J=ceil(0.28466*P-0.28151):Jhigh=floor(0.28466*P-0.11695)
30 if J<>Jhigh then 100
40 D=2^(P-2)*5^J:C=5^(P-J)
50 B=C+D:A=C-D:N=A*A:M=B*B
60 print "P=";P;"J=";J;"D=";D;"C=";C
70 print "A=";A;"B=";B:print "N=";N:print "M=";M
80 print
100 K=ceil(0.71534*P-1.14286):Khigh=floor(0.71534*P-0.97830)
110 if K<>Khigh then 200
120 D=5^K:C=2^(P-2)*5^(P-K)
130 B=C+D:A=C-D:N=A*A:M=B*B
140 print "P=";P;"K=";K;"D=";D;"C=";C
150 print "A=";A;"B=";B:print "N=";N:print "M=";M
190 print
200 next P
210 print=print
220 end
The output:
P= 4 J= 1 D= 20 C= 125
A= 105 B= 145
N= 11025
M= 21025
P= 7 K= 4 D= 625 C= 4000
A= 3375 B= 4625
N= 11390625
M= 21390625
P= 8 J= 2 D= 1600 C= 15625
A= 14025 B= 17225
N= 196700625
M= 296700625
P= 11 J= 3 D= 64000 C= 390625
A= 326625 B= 454625
N= 106683890625
M= 206683890625
P= 14 K= 9 D= 1953125 C= 12800000
A= 10846875 B= 14753125
N= 117654697265625
M= 217654697265625
P= 15 J= 4 D= 5120000 C= 48828125
A= 43708125 B= 53948125
N= 1910400191015625
M= 2910400191015625
P= 18 J= 5 D= 204800000 C= 1220703125
A= 1015903125 B= 1425503125
N= 1032059159384765625
M= 2032059159384765625
P= 21 K= 14 D= 6103515625 C= 40960000000
A= 34856484375 B= 47063515625
N= 1214974502984619140625
M= 2214974502984619140625
P= 22 J= 6 D= 16384000000 C= 152587890625
A= 136203890625 B= 168971890625
N= 18551499821386962890625
M= 28551499821386962890625
P= 28 K= 19 D= 19073486328125 C= 131072000000000
A= 111998513671875 B= 150145486328125
N= 12543667064709171295166015625
M= 22543667064709171295166015625
P= 29 J= 8 D= 52428800000000 C= 476837158203125
A= 424408358203125 B= 529265958203125
N= 180122454512672059478759765625
M= 280122454512672059478759765625
Edited on April 11, 2022, 12:22 pm
Going much further
