## 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.

math grad - Interest: Number theory
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### 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.

• benvitalis says:

Nice! … I usually ignore the leading zeros. I was just curious this time.