Concatenation : a || b = n^2 + a = (n+1)^2 – b

 
 
Notice that
122 \; = \; 1 \; || \; 22 \; = \; 11^2 \; + \; 1 \; = \; 12^2 \; - \; 22

The integer N = 122 can be partitioned into two parts, 1 and 22, so that
the first part is the difference between N and the greatest square less than N, and
the second part is the difference between N and the least square greater than N

 

10202 \; = \; 101^2 \; + \; 1 \; = \; 102^2 \; - \; 0202
1002002 \; = \; 1001^2 \; + \; 1 \; = \; 1002^2 \; - \; 002002
100020002 \; = \; 10001^2 \; + \; 1 \; = \; 10002^2 \; - \; 00020002

 

other numbers with this property,

35322 \; = \; 353 \; || \; 22 \; = \; 187^2 \; + \; 353 \; = \; 188^2 \; - \; 22

180125042 \; = \; 1801 \; || \; 25042
  = \; 13421^2 \; + \; 1801 \; = \; 13422^2 \; - \; 25042

395930202 \; = \; 39593 \; || \; 0202
  = \; 19897^2 \; + \; 39593 \; = \; 19898^2 \; - \; 202

34811325042 \; = \; 348113 \; || \; 25042
  = \; 186577^2 \; + \; 348113 \; = \; 186578^2 \; - \; 25042

12863998200 \; = \; 128639 \; || \; 98200
  = \; 113419^2 \; + \; 128639 \; = \; 113420^2 \; - \; 98200

3995993002002 \; = \; 3995993 \; || \; 002002
  = \; 1998997^2 \; + \; 3995993 \; = \; 1998998^2 \; - \; 002002

3259649351250 \; = \; 3259649 \; || \; 351250
  = \; 1805449^2 \; + \; 3259649 \; = \; 1805450^2 \; - \; 351250

1426943962152 \; = \; 1426943 \; || \; 962152
  = \; 1194547^2 \; + \; 1426943 \; = \; 1194548^2 \; - \; 962152

251673596561088 \; = \; 25167359 \; || \; 6561088
  = \; 15864223^2 \; + \; 25167359 \; = \; 15864224^2 \; - \; 6561088

199820098289538 \; = \; 19982009 \; || \; 8289538
  = \; 14135773^2 \; + \; 19982009 \; = \; 14135774^2 \; - \; 8289538

39995999300020002 \; = \; 399959993 \; || \; 00020002
  = \; 199989997^2 \; + \; 399959993 \; = \; 199989998^2 \; - \; 00020002

31995187337792098 \; = \; 319951873 \; || \; 37792098
  = \; 178871985^2 \; + \; 319951873 \; = \; 178871986^2 \; - \; 37792098

31780502338736712 \; = \; 317805023 \; || \; 38736712
  = \; 178270867^2 \; + \; 317805023 \; = \; 178270868^2 \; - \; 38736712

24690734367358368 \; = \; 246907343 \; || \; 67358368
  = \; 157132855^2 \; + \; 246907343 \; = \; 157132856^2 \; - \; 67358368

20411020181624082 \; = \; 204110201 \; || \; 81624082
  = \; 142867141^2 \; + \; 204110201 \; = \; 142867142^2 \; - \; 81624082

14817980995278450  \;= \; 148179809 \; || \; 95278450
  = \; 121729129^2 \; + \; 148179809 \; = \; 121729130^2 \; - \; 95278450

14671995195536072 \; = \; 146719951 \; || \; 95536072
  = \; 121128011^2 \; + \; 146719951 \; = \; 121128012^2 \; - \; 95536072

6220790395536072 \; = \; 62207903 \; || \; 95536072
  = \; 78871987^2 \; + \; 62207903 \; = \; 78871988^2 \; - \; 95536072

6126328995278450 \; = \; 61263289 \; || \; 95278450
  = \; 78270869^2 \; + \; 61263289 \; = \; 78270870^2 \; - \; 95278450

3264163381624082 \; = \; 32641633 \; || \; 81624082
  = \; 57132857^2 \; + \; 32641633 \; = \; 57132858^2 \; - \; 81624082

1837591967358368 \; = \; 18375919 \; || \; 67358368
  = \; 42867143^2 \; + \; 18375919 \; = \; 42867144^2 \; - \; 67358368

472155138736712 \; = \; 4721551 \; || \; 38736712
  = \; 21729131^2 \; + \; 4721551 \; = \; 21729132^2 \; - \; 38736712

446392937792098 \; = \; 4463929 \; || \; 37792098
  = \; 21128013^2 \; + \; 4463929 \; = \; 21128014^2 \; - \; 37792098

…………………………………
…………………………………

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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