Concatenation: aa||bb||cc||dd + 1, aaa||bbb + 1 are squares

 
 

Numbers of the form   aaa \; || \; bbb   such that   aaa \; || \; bbb + 1   is a square :
 

22200 \; + \; 1 \; = \; 149^2
 

111 \; || \; 555 \; + \; 1 \; = \; 334^2
1111 \; || \; 5555 \; + \; 1 \; = \; 3334^2
11111 \; || \; 55555 \; + \; 1 \; = \; 33334^2
……………………..
……………………..

444 \; || \; 888 \; + \; 1 \; = \; 667^2
4444 \; || \; 8888 \; + \; 1 \; = \; 6667^2
44444 \; || \; 88888 \; + \; 1 \; = \; 66667^2
………………………….
………………………….

7^2 \; - \; 4^2 \; = \; 33
67^2 \; - \; 34^2 \; = \; 3333
667^2 \; - \; 334^2 \; = \; 333333
6667^2 \; - \; 3334^2 \; = \; 33333333
……………………….
……………………….

 

Numbers of the form   aa \; || \; bb \; || \; cc \; || \; dd + 1   is a square :

11 \; || \; 33 \; || \; 66 \; || \; 88 \; + \; 1 \; = \; 3367^2
111 \; || \; 333 \; || \; 666 \; || \; 888 \; + \; 1 \; = \; 333667^2
1111 \; || \; 3333 \; || \; 6666 \; || \; 8888 \; + \; 1 \; = \; 33336667^2
11111 \; || \; 33333 \; || \; 66666 \; || \; 88888 \; + \; 1 \; = \; 3333366667^2
…………………………….
…………………………….

 

22886655 \; + \; 1 \; = \; 4784^2

 

44 \; || \; 00 \; || \; 99 \; || \; 55 \; + \; 1 \; = \; 6634^2
444 \; || \; 000 \; || \; 999 \; || \; 555 \; + \; 1 \; = \; 666334^2
4444 \; || \; 0000 \; || \; 9999 \; || \; 5555 \; + \; 1 \; = \; 66663334^2
44444 \; || \; 00000 \; || \; 99999 \; || \; 55555 \; + \; 1 \; = \; 6666633334^2

634^2 \; - \; 367^2 \; = \; 267 \; \times \; 1001
6634^2 \; - \; 3367^2 \; = \; 3267 \; \times \; 10001
66634^2 \; - \; 33367^2 \; = \; 33267 \; \times \; 100001
666634^2 \; - \; 333367^2 \; = \; 333267 \; \times \; 1000001
…………………………………
…………………………………

 

Can you find other examples?

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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