Num3er 11296321

$11296321 \; = \; 3361^2$

$11296321$   is a perfect square containing precisely one 3, two 2’s and three 1’s.

Any of the other seven digits   (0, 4, 5, 6, 7, 8, 9)   occur only once.

Find other examples.

Paul found:

$3483^2 \; = \; 12131289$
$3596^2 \; = \; 12931216$
$3623^2 \; = \; 13126129$
$3889^2 \; = \; 15124321$

$4649^2 \; = \; 21613201$

$5489^2 \; = \; 30129121$
$5579^2 \; = \; 31125241$

$10642^2 \; = \; 113252164$
$10915^2 \; = \; 119137225$
$10961^2 \; = \; 120143521$
$11059^2 \; = \; 122301481$
$11127^2 \; = \; 123810129$
$11189^2 \; = \; 125193721$
$11455^2 \; = \; 131217025$
$11477^2 \; = \; 131721529$
$11965^2 \; = \; 143161225$
$12579^2 \; = \; 158231241$
$12761^2 \; = \; 162843121$
$13127^2 \; = \; 172318129$
$13131^2 \; = \; 172423161$
$13421^2 \; = \; 180123241$
$13899^2 \; = \; 193182201$
$14534^2 \; = \; 211237156$
$14601^2 \; = \; 213189201$
$14711^2 \; = \; 216413521$
$15944^2 \; = \; 254211136$
$16761^2 \; = \; 280931121$
$17239^2 \; = \; 297183121$
$17373^2 \; = \; 301821129$
$17739^2 \; = \; 314672121$
$19285^2 \; = \; 371911225$
$19489^2 \; = \; 379821121$

$20546^2 \; = \; 422138116$
$22829^2 \; = \; 521163241$
$27261^2 \; = \; 743162121$

$30185^2 \; = \; 911134225$
$30351^2 \; = \; 921183201$
$31817^2 \; = \; 1012321489$
$31818^2 \; = \; 1012385124$
$32591^2 \; = \; 1062173281$
$33425^2 \; = \; 1117230625$
$33455^2 \; = \; 1119237025$
$33531^2 \; = \; 1124327961$
$33543^2 \; = \; 1125132849$
$33561^2 \; = \; 1126340721$
$33573^2 \; = \; 1127146329$
$33693^2 \; = \; 1135218249$
$33715^2 \; = \; 1136701225$
$33885^2 \; = \; 1148193225$
$34019^2 \; = \; 1157292361$
$34211^2 \; = \; 1170392521$
$34572^2 \; = \; 1195223184$
$34661^2 \; = \; 1201384921$
$34696^2 \; = \; 1203812416$
$34732^2 \; = \; 1206311824$
$34829^2 \; = \; 1213059241$
$34894^2 \; = \; 1217591236$
$35033^2 \; = \; 1227311089$
$35034^2 \; = \; 1227381156$
$35211^2 \; = \; 1239814521$
$35601^2 \; = \; 1267431201$
$35709^2 \; = \; 1275132681$
$35769^2 \; = \; 1279421361$
$35794^2 \; = \; 1281210436$
$35921^2 \; = \; 1290318241$
$36089^2 \; = \; 1302415921$
$36223^2 \; = \; 1312105729$
$36349^2 \; = \; 1321249801$
$36369^2 \; = \; 1322704161$
$36759^2 \; = \; 1351224081$
$36895^2 \; = \; 1361241025$
$37182^2 \; = \; 1382501124$
$37285^2 \; = \; 1390171225$
$37761^2 \; = \; 1425893121$

$40267^2 \; = \; 1621431289$
$40523^2 \; = \; 1642113529$
$40649^2 \; = \; 1652341201$
$40965^2 \; = \; 1678131225$
$41139^2 \; = \; 1692417321$
$41246^2 \; = \; 1701232516$
$41365^2 \; = \; 1711063225$
$41514^2 \; = \; 1723412196$
$42454^2 \; = \; 1802342116$
$42606^2 \; = \; 1815271236$
$42682^2 \; = \; 1821753124$
$42698^2 \; = \; 1823119204$
$42795^2 \; = \; 1831412025$
$43511^2 \; = \; 1893207121$
$43718^2 \; = \; 1911263524$
$44139^2 \; = \; 1948251321$
$44939^2 \; = \; 2019513721$
$44957^2 \; = \; 2021131849$
$45069^2 \; = \; 2031214761$
$45841^2 \; = \; 2101397281$
$45966^2 \; = \; 2112873156$
$45973^2 \; = \; 2113516729$
$46209^2 \; = \; 2135271681$
$46239^2 \; = \; 2138045121$
$46601^2 \; = \; 2171653201$
$46604^2 \; = \; 2171932816$
$46639^2 \; = \; 2175196321$
$46831^2 \; = \; 2193142561$
$47646^2 \; = \; 2270141316$
$47971^2 \; = \; 2301216841$
$48079^2 \; = \; 2311590241$
$48081^2 \; = \; 2311782561$
$48178^2 \; = \; 2321119684$
$48381^2 \; = \; 2340721161$
$49111^2 \; = \; 2411890321$
$49711^2 \; = \; 2471183521$
$49811^2 \; = \; 2481135721$
$49861^2 \; = \; 2486119321$

$50113^2 \; = \; 2511312769$
$50118^2 \; = \; 2511813924$
$50181^2 \; = \; 2518132761$
$50311^2 \; = \; 2531196721$
$50318^2 \; = \; 2531901124$
$51139^2 \; = \; 2615197321$
$52263^2 \; = \; 2731421169$
$53211^2 \; = \; 2831410521$
$53489^2 \; = \; 2861073121$
$54509^2 \; = \; 2971231081$
$54911^2 \; = \; 3015217921$
$55689^2 \; = \; 3101264721$
$55715^2 \; = \; 3104161225$
$55901^2 \; = \; 3124921801$
$55921^2 \; = \; 3127158241$
$57159^2 \; = \; 3267151281$
$57198^2 \; = \; 3271611204$
$58404^2 \; = \; 3411027216$

$60179^2 \; = \; 3621512041$
$61261^2 \; = \; 3752910121$
$64623^2 \; = \; 4176132129$
$64895^2 \; = \; 4211361025$
$64901^2 \; = \; 4212139801$
$67239^2 \; = \; 4521083121$
$69373^2 \; = \; 4812613129$

$70115^2 \; = \; 4916113225$
$71604^2 \; = \; 5127132816$
$72239^2 \; = \; 5218473121$
$72761^2 \; = \; 5294163121$

$83239^2 \; = \; 6928731121$
$84611^2 \; = \; 7159021321$
$87739^2 \; = \; 7698132121$
$88389^2 \; = \; 7812615321$
$88945^2 \; = \; 7911213025$

$94611^2 \; = \; 8951241321$
$95039^2 \; = \; 9032411521$
$95482^2 \; = \; 9116812324$
$95559^2 \; = \; 9131522481$

math grad - Interest: Number theory
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2 Responses to Num3er 11296321

1. Paul says:

There seem to be quite a few, here are those with n<=100000, format n, n^2.

3361 , 11296321
3483 , 12131289
3596 , 12931216
3623 , 13126129
3889 , 15124321
4649 , 21613201
5489 , 30129121
5579 , 31125241
10642 , 113252164
10915 , 119137225
10961 , 120143521
11059 , 122301481
11127 , 123810129
11189 , 125193721
11455 , 131217025
11477 , 131721529
11965 , 143161225
12579 , 158231241
12761 , 162843121
13127 , 172318129
13131 , 172423161
13421 , 180123241
13899 , 193182201
14534 , 211237156
14601 , 213189201
14711 , 216413521
15944 , 254211136
16761 , 280931121
17239 , 297183121
17373 , 301821129
17739 , 314672121
19285 , 371911225
19489 , 379821121
20546 , 422138116
22829 , 521163241
27261 , 743162121
30185 , 911134225
30351 , 921183201
31817 , 1012321489
31818 , 1012385124
32591 , 1062173281
33425 , 1117230625
33455 , 1119237025
33531 , 1124327961
33543 , 1125132849
33561 , 1126340721
33573 , 1127146329
33693 , 1135218249
33715 , 1136701225
33885 , 1148193225
34019 , 1157292361
34211 , 1170392521
34572 , 1195223184
34661 , 1201384921
34696 , 1203812416
34732 , 1206311824
34829 , 1213059241
34894 , 1217591236
35033 , 1227311089
35034 , 1227381156
35211 , 1239814521
35601 , 1267431201
35709 , 1275132681
35769 , 1279421361
35794 , 1281210436
35921 , 1290318241
36089 , 1302415921
36223 , 1312105729
36349 , 1321249801
36369 , 1322704161
36759 , 1351224081
36895 , 1361241025
37182 , 1382501124
37285 , 1390171225
37761 , 1425893121
40267 , 1621431289
40523 , 1642113529
40649 , 1652341201
40965 , 1678131225
41139 , 1692417321
41246 , 1701232516
41365 , 1711063225
41514 , 1723412196
42454 , 1802342116
42606 , 1815271236
42682 , 1821753124
42698 , 1823119204
42795 , 1831412025
43511 , 1893207121
43718 , 1911263524
44139 , 1948251321
44939 , 2019513721
44957 , 2021131849
45069 , 2031214761
45841 , 2101397281
45966 , 2112873156
45973 , 2113516729
46209 , 2135271681
46239 , 2138045121
46601 , 2171653201
46604 , 2171932816
46639 , 2175196321
46831 , 2193142561
47646 , 2270141316
47971 , 2301216841
48079 , 2311590241
48081 , 2311782561
48178 , 2321119684
48381 , 2340721161
49111 , 2411890321
49711 , 2471183521
49811 , 2481135721
49861 , 2486119321
50113 , 2511312769
50118 , 2511813924
50181 , 2518132761
50311 , 2531196721
50318 , 2531901124
51139 , 2615197321
52263 , 2731421169
53211 , 2831410521
53489 , 2861073121
54509 , 2971231081
54911 , 3015217921
55689 , 3101264721
55715 , 3104161225
55901 , 3124921801
55921 , 3127158241
57159 , 3267151281
57198 , 3271611204
58404 , 3411027216
60179 , 3621512041
61261 , 3752910121
64623 , 4176132129
64895 , 4211361025
64901 , 4212139801
67239 , 4521083121
69373 , 4812613129
70115 , 4916113225
71604 , 5127132816
72239 , 5218473121
72761 , 5294163121
83239 , 6928731121
84611 , 7159021321
87739 , 7698132121
88389 , 7812615321
88945 , 7911213025
94611 , 8951241321
95039 , 9032411521
95482 , 9116812324
95559 , 9131522481

Paul

• benvitalis says:

Nice! Thanks for your contribution