Concatenation : A^2 || B^2 = C^2, (A=10,11, …, 19)

 

A^2 \; || \; B^2 \; = \; C^2,         with    A \; = \; 10, \; 11, \; \dotsm \; , \; 19
 

                                                               A \; = \; 10
 

10^2 \; || \; B^2 \; = \; C^2
 

Jeff Curtis found:

1001125158203125^2
= 1002251582387232059478759765625
= 100 \; || \; 2251582387232059478759765625
= 10^2 \; || \; 47450841796875^2

 
 
 

                                                             ——————————————
 
 
 

                                                               A \; = \; 11
 
347875^2 \; = \; 121017015625 \; = \; 121 \; || \; 017015625 \; = \; 11^2 \; || \; 4125^2

 
 

Paul found:

3490625^2 = 12184462890625 = 121 \; || \; 84462890625 = 11^2 \;  || \; 290625^2

 
 

                                                             ——————————————
 
 
 

                                                               A \; = \; 12
 

38^2 \; = \; 1444 \; = \; 144 \; || \; 4 \; = \; 12^2 \; || \; 2^2

380^2 \; = \; 1444 \; = \; 144 \; || \; 400 \; = \; 12^2 \; || \; 20^2

1201^2 \; = \; 1442401 \; = \; 144 \; || \; 2401 \; = \; 12^2 \; || \; 49^2

 

Paul Found:

3800^2 = 14440000 = 144 \; || \; 40000 = 12^2 \; || \; 200^2
12010^2 = 144240100 = 144 \; || \; 240100 = 12^2 \; || \; 490^2
12025^2 = 144600625 = 144 \; || \; 600625 = 12^2 \; || \; 775^2
38000^2 = 1444000000 = 144 \; || \; 4000000 = 12^2 \; || \; 2000^2
120100^2 = 14424010000 = 144 \; || \; 24010000 = 12^2 \; || \; 4900^2
120250^2 = 14460062500 = 144 \; || \; 60062500 = 12^2 \; || \; 7750^2
380000^2 = 144400000000 = 144 \; || \; 400000000 = 12^2 \; || \; 20000^2
1201000^2 = 1442401000000 = 144 \; || \; 2401000000 = 12^2 \; || \; 49000^2
1202500^2 = 1446006250000 = 144 \; || \; 6006250000 = 12^2 \; || \; 77500^2
3796325^2 = 14412083505625 = 144 \; || \; 12083505625 = 12^2 \; || \; 109925^2

 
 
 

                                                             ——————————————
 
 

                                                               A \; = \; 13
 

1301^2 \; = \; 1692601 \; = \; 169 \; || \; 2601 \; = \; 13^2 \; || \; 51^2

 
 

Paul found:

13010^2 = 169260100 = 169 \; || \; 260100 = 13^2 \; || \; 510^2
41125^2 = 1691265625 = 169 \; || \; 1265625 = 13^2 \; || \; 1125^2
130100^2 = 16926010000 = 169 \; || \; 26010000 = 13^2 \; || \; 5100^2
411250^2 = 169126562500 = 169 \; || \; 126562500 = 13^2 \; || \; 11250^2
1301000^2 = 1692601000000 = 169 \; || \; 2601000000 = 13^2 \; || \; 51000^2
4112500^2 = 16912656250000 = 169 \; || \; 12656250000 = 13^2 \; || \; 112500^2
4116325^2 = 16944131505625 = 169 \; || \; 44131505625 = 13^2 \; || \; 210075^2

 

                                                             ——————————————
 
 

                                                               A \; = \; 14
 

44275^2 \;  = \; 1960275625 \;  = \; 196 \;  || \;  0275625 \;  = \; 14^2 \;  || \; 525^2

 
 

                                                             ——————————————
 
 

                                                               A \; = \; 15
 

475^2 \;  = \;  225625 \;  = \;  225 \;  || \;  625 \;  = \;  15^2 \;  || \;  25^2

 
 

Paul found:

4750^2 = 22562500 = 225 \; || \; 62500 = 15^2 \; || \; 250^2
47500^2 = 2256250000 = 225 \; || \; 6250000 = 15^2 \; || \; 2500^2
150125^2 = 22537515625 = 225 \; || \; 37515625 = 15^2 \; || \; 6125^2
475000^2 = 225625000000 = 225 \; || \; 625000000 = 15^2 \; || \; 25000^2
1501250^2 = 2253751562500 = 225 \; || \; 3751562500 = 15^2 \; || \; 61250^2
1503125^2 = 2259384765625 = 225 \; || \; 9384765625 = 15^2 \; || \; 96875^2
4750000^2 = 22562500000000 = 225 \; || \; 62500000000 = 15^2 \; || \; 250000^2

 

                                                             ——————————————
 
 

                                                               A \; = \; 16
 

5060^2 \;  = \; 25603600 \;  = \; 256 \;  || \;  03600 \;  = \; 16^2 \;  || \; 60^2

160045^2 \;  = \; 25614402025 \;  = \; 256 \;  || \;  14402025 \;  = \; 16^2 \;  || \; 3795^2

Note that:     3795^2 \;  = \; 144 \;  || \;  02025 \;  = \; 12^2 \;  || \; 45^2

 
 

Paul found:

1600450^2 = 2561440202500 = 256 \; || \; 1440202500 = 16^2 \; || \; 37950^2
16004500^2 = 256144020250000 = 256 \; || \; 144020250000 = 16^2 \; || \; 379500^2
 

                                                             ——————————————
 
 

                                                               A \; = \; 17
 

537625^2 \;  = \; 289040640625 \;  = \; 289 \;  || \;  040640625 \;  = \; 17^2 \;  || \; 6375^2

 
 

                                                             ——————————————
 
 

                                                               A \; = \; 18
 

570^2 \;  = \; 324900 \;  = \; 324 \;  || \;  900 \;  = \; 18^2 \;  || \; 30^2

 
 
Paul found:

57^2 = 3249 = 324 \; || \; 9 = 18^2 \; || \; 3^2
5700^2 = 32490000 = 324 \; || \; 90000= 18^2 \; || \; 300^2
18015^2 = 324540225 = 324 \; || \; 540225 = 18^2 \; || \; 735^2
57000^2 = 3249000000 = 324 \; || \; 9000000 = 18^2 \; || \; 3000^2
180150^2 = 32454022500 = 324 \; || \; 54022500 = 18^2 \; || \; 7350^2
570000^2 = 324900000000 = 324 \; || \; 900000000 = 18^2 \; || \; 30000^2
1801500^2 = 3245402250000 = 324 \; || \; 5402250000 = 18^2 \; || \; 73500^2

 

                                                             ——————————————
 
 

                                                               A \; = \; 19
 

6013^2 \;  = \; 36156169 \;  = \; 361 \;  || \;  56169 \;  = \; 19^2 \;  || \;  237^2
 

Paul found:

19025^2 = 361950625 = 361||950625 = 19^2 \;  || \;   975^2
60130^2 = 3615616900 = 361||5616900 = 19^2 \;  || \;   2370^2
190250^2 = 36195062500 = 361||95062500 = 19^2 \;  || \;   9750^2
601300^2 = 361561690000 = 361||561690000 = 19^2 \;  || \;   23700^2
1902500^2 = 3619506250000 = 361||9506250000 = 19^2 \;  || \;   97500^2
6013000^2 = 36156169000000 = 361||56169000000 = 19^2 \;  || \;   237000^2
19007225^2 = 361274602200625 = 361||274602200625 = 19^2 \;  || \;   524025^2
19025000^2 = 361950625000000 = 361||950625000000 = 19^2 \;  || \;   975000^2

 
 

Find more solutions.
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

3 Responses to Concatenation : A^2 || B^2 = C^2, (A=10,11, …, 19)

  1. Paul says:

    I can’t say I am a fan of using leading zeros to form the concatenation but hey ho:)

    here are a few other solutions

    11^2||290625^2, = 3490625^2

    12^2||200^2, = 3800^2
    12^2||490^2, = 12010^2
    12^2||775^2, = 12025^2
    12^2||2000^2, = 38000^2
    12^2||4900^2, = 120100^2
    12^2||7750^2, = 120250^2
    12^2||20000^2, = 380000^2
    12^2||49000^2, = 1201000^2
    12^2||77500^2, = 1202500^2
    12^2||109925^2, = 3796325^2

    13^2||510^2, = 13010^2
    13^2||1125^2, = 41125^2
    13^2||5100^2, = 130100^2
    13^2||11250^2, = 411250^2
    13^2||51000^2, = 1301000^2
    13^2||112500^2, = 4112500^2
    13^2||210075^2, = 4116325^2

    15^2||250^2, = 4750^2
    15^2||2500^2, = 47500^2
    15^2||6125^2, = 150125^2
    15^2||25000^2, = 475000^2
    15^2||61250^2, = 1501250^2
    15^2||96875^2, = 1503125^2
    15^2||250000^2, = 4750000^2

    16^2||37950^2, = 1600450^2
    16^2||379500^2, = 16004500^2

    18^2||3^2, = 57^2
    18^2||300^2, = 5700^2
    18^2||735^2, = 18015^2
    18^2||3000^2, = 57000^2
    18^2||7350^2, = 180150^2
    18^2||30000^2, = 570000^2
    18^2||73500^2, = 1801500^2

    19^2||975^2, = 19025^2
    19^2||2370^2, = 60130^2
    19^2||9750^2, = 190250^2
    19^2||23700^2, = 601300^2
    19^2||97500^2, = 1902500^2
    19^2||237000^2, = 6013000^2
    19^2||524025^2, = 19007225^2
    19^2||975000^2, = 19025000^2

    Paul.

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