## 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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