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 N & n (Posted on 2014-01-27)
N is a m-th power of an integer.
It consists of n distinct digits, averaging n.

Find n,m and N, using your head + pen and paper only.

Check, whether there exist additional solutions (computer allowed).

Clearly, n>1; m>1 - to exclude trivial cases.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 98 results not requiring "distinct" Comment 7 of 7 |

144       =   12 ^ 2
216       =   6 ^ 3
225       =   15 ^ 2
243       =   3 ^ 5
324       =   18 ^ 2
441       =   21 ^ 2
900       =   30 ^ 2
1681      =   41 ^ 2
3364      =   58 ^ 2
3481      =   59 ^ 2
4624      =   68 ^ 2
7225      =   85 ^ 2
9025      =   95 ^ 2
12769     =   113 ^ 2
14884     =   122 ^ 2
24649     =   157 ^ 2
24964     =   158 ^ 2
27556     =   166 ^ 2
30976     =   176 ^ 2
33856     =   184 ^ 2
37249     =   193 ^ 2
37636     =   194 ^ 2
44944     =   212 ^ 2
48841     =   221 ^ 2
56644     =   238 ^ 2
65536     =   256 ^ 2
65536     =   16 ^ 4
65536     =   4 ^ 8
65536     =   2 ^ 16
66049     =   257 ^ 2
70756     =   266 ^ 2
75076     =   274 ^ 2
75625     =   275 ^ 2
80089     =   283 ^ 2
80656     =   284 ^ 2
85264     =   292 ^ 2
96721     =   311 ^ 2
149769    =   387 ^ 2
173889    =   417 ^ 2
178929    =   423 ^ 2
199809    =   447 ^ 2
278784    =   528 ^ 2
279936    =   6 ^ 7
287496    =   66 ^ 3
288369    =   537 ^ 2
294849    =   543 ^ 2
389376    =   624 ^ 2
439569    =   663 ^ 2
459684    =   678 ^ 2
467856    =   684 ^ 2
471969    =   687 ^ 2
509796    =   714 ^ 2
589824    =   768 ^ 2
599076    =   774 ^ 2
617796    =   786 ^ 2
660969    =   813 ^ 2
665856    =   816 ^ 2
675684    =   822 ^ 2
685584    =   828 ^ 2
695556    =   834 ^ 2
746496    =   864 ^ 2
751689    =   867 ^ 2
759375    =   15 ^ 5
767376    =   876 ^ 2
777924    =   882 ^ 2
788544    =   888 ^ 2
793881    =   891 ^ 2
799236    =   894 ^ 2
853776    =   924 ^ 2
859329    =   927 ^ 2
870489    =   933 ^ 2
876096    =   936 ^ 2
884736    =   96 ^ 3
887364    =   942 ^ 2
898704    =   948 ^ 2
915849    =   957 ^ 2
927369    =   963 ^ 2
938961    =   969 ^ 2
944784    =   972 ^ 2
956484    =   978 ^ 2
968256    =   984 ^ 2
970299    =   99 ^ 3
974169    =   987 ^ 2
986049    =   993 ^ 2
2778889   =   1667 ^ 2
4695889   =   2167 ^ 2
5678689   =   2383 ^ 2
5697769   =   2387 ^ 2
5938969   =   2437 ^ 2
6568969   =   2563 ^ 2
6589489   =   2567 ^ 2
6848689   =   2617 ^ 2
6895876   =   2626 ^ 2
7974976   =   2824 ^ 2
7997584   =   2828 ^ 2
8779369   =   2963 ^ 2
9878449   =   3143 ^ 2
9966649   =   3157 ^ 2

Indeed there are no 8-digit results and of course no 9-digit.

 Posted by Charlie on 2014-01-27 15:48:49

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