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Neither 3 nor 6 (Posted on 2011-03-11) Difficulty: 4 of 5
0, 25, 2025, 13225…are squares that remain squares if every digit in the number defining them is augmented by 1.
Let's call them squarish numbers.

a. List two more samples of squarish numbers.
b. Prove that all such numbers are evenly divisible by 25.
c. Why are there neither 3-digit nor 6-digit squarish numbers?
d. Prove that between 10^k and 10^(k+1) there is at most one squarish number.

See The Solution Submitted by Ady TZIDON    
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re: Part a only (spoiler) | Comment 3 of 6 |
(In reply to Part a only (spoiler) by Charlie)

Sloane's OEIS lists

A061843 as a simple table:
n    a(n)
1    0
2    25
3    2025
4    13225
5    4862025
6    60415182025
7    207612366025
8    153668543313582025
9    13876266042653742025
10   20761288044852366025
11   47285734107144405625
12   406066810454367265225
13   141704161680410868660551655625


 

Edited on March 11, 2011, 4:34 pm
  Posted by Charlie on 2011-03-11 16:32:42

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