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Repeating squares (Posted on 2020-09-09) Difficulty: 3 of 5
A repeating number is a number such that the first half of the digits are the same as the second half of the digits. For example, 2020 is a repeating number. What is the smallest repeating number that is also a square?

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re(3): Solution Comment 5 of 5 |
(In reply to re(2): Solution by Larry)

Looking again at Sloan's A092118, the 8th number in the series has 42 digits, so "n" being half the number of digits is 21 corresponding to one of its factors being 100..001 where there are 20 zeros between the 1's.
It turns out this 100..001 is divisible by 7^2.

I checked values of n up to 100 and values of prime numbers (p) up to 271.

 n  p
11 11  (meaning 121 is a factor of 100000000001)
21  7
33 11
39 13
55 11
63  7
77 11
99 11
  Posted by Larry on 2020-09-16 12:21:21

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