By adding 1 to the positive base N integer having the form XXXYYY with non leading zeroes, we get a perfect square- where X and Y are not necessarily distinct.
What are the value(s) of N, with 2 ≤ N ≤ 16, for which this is possible?
The simple, D2 answer to the question, is all N. For any N, if X=Y=N-1, then XXXYYY will be N^6 - 1 and adding 1 will result in N^6, a perfect square.
Here's a computer printout (going beyond the asked-for N=16):
(each repetition of X or Y is represented in decimal for those digits larger than 9)
N X X X Y Y Y XXXYYY (dec.) +1 sq root
2 1 1 1 1 1 1 63 64 8
3 2 2 2 2 2 2 728 729 27
4 3 3 3 3 3 3 4095 4096 64
5 1 1 1 3 3 3 3968 3969 63
5 4 4 4 4 4 4 15624 15625 125
6 5 5 5 5 5 5 46655 46656 216
7 6 6 6 6 6 6 117648 117649 343
8 7 7 7 7 7 7 262143 262144 512
9 2 2 2 6 6 6 133224 133225 365
9 8 8 8 8 8 8 531440 531441 729
10 1 1 1 5 5 5 111555 111556 334
10 4 4 4 8 8 8 444888 444889 667
10 9 9 9 9 9 9 999999 1000000 1000
11 10 10 10 10 10 10 1771560 1771561 1331
12 11 11 11 11 11 11 2985983 2985984 1728
13 3 3 3 9 9 9 1207800 1207801 1099
13 12 12 12 12 12 12 4826808 4826809 2197
14 13 13 13 13 13 13 7529535 7529536 2744
15 14 14 14 14 14 14 11390624 11390625 3375
16 5 5 5 8 8 8 5593224 5593225 2365
16 15 15 15 15 15 15 16777215 16777216 4096
17 1 1 1 7 7 7 1510440 1510441 1229
17 4 4 4 12 12 12 6036848 6036849 2457
17 9 9 9 15 15 15 13579224 13579225 3685
17 16 16 16 16 16 16 24137568 24137569 4913
18 17 17 17 17 17 17 34012223 34012224 5832
19 2 2 2 10 10 10 5230368 5230369 2287
19 8 8 8 16 16 16 20912328 20912329 4573
19 18 18 18 18 18 18 47045880 47045881 6859
20 19 19 19 19 19 19 63999999 64000000 8000
21 5 5 5 15 15 15 21446160 21446161 4631
21 20 20 20 20 20 20 85766120 85766121 9261
22 21 21 21 21 21 21 113379903 113379904 10648
23 22 22 22 22 22 22 148035888 148035889 12167
24 23 23 23 23 23 23 191102975 191102976 13824
25 6 6 6 18 18 18 61042968 61042969 7813
25 13 13 13 24 24 24 132249999 132250000 11500
25 24 24 24 24 24 24 244140624 244140625 15625
26 1 1 1 9 9 9 12362255 12362256 3516
26 4 4 4 16 16 16 49434960 49434961 7031
26 9 9 9 21 21 21 111218115 111218116 10546
26 16 16 16 24 24 24 197711720 197711721 14061
26 25 25 25 25 25 25 308915775 308915776 17576
27 26 26 26 26 26 26 387420488 387420489 19683
28 3 3 3 15 15 15 53553123 53553124 7318
28 12 12 12 24 24 24 214183224 214183225 14635
28 27 27 27 27 27 27 481890303 481890304 21952
More varied would be only those cases where X is not the same as Y:
N X X X Y Y Y XXXYYY (dec.) +1 sq root
5 1 1 1 3 3 3 3968 3969 63
9 2 2 2 6 6 6 133224 133225 365
10 1 1 1 5 5 5 111555 111556 334
10 4 4 4 8 8 8 444888 444889 667
13 3 3 3 9 9 9 1207800 1207801 1099
16 5 5 5 8 8 8 5593224 5593225 2365
17 1 1 1 7 7 7 1510440 1510441 1229
17 4 4 4 12 12 12 6036848 6036849 2457
17 9 9 9 15 15 15 13579224 13579225 3685
19 2 2 2 10 10 10 5230368 5230369 2287
19 8 8 8 16 16 16 20912328 20912329 4573
21 5 5 5 15 15 15 21446160 21446161 4631
25 6 6 6 18 18 18 61042968 61042969 7813
25 13 13 13 24 24 24 132249999 132250000 11500
26 1 1 1 9 9 9 12362255 12362256 3516
26 4 4 4 16 16 16 49434960 49434961 7031
26 9 9 9 21 21 21 111218115 111218116 10546
26 16 16 16 24 24 24 197711720 197711721 14061
28 3 3 3 15 15 15 53553123 53553124 7318
28 12 12 12 24 24 24 214183224 214183225 14635
29 7 7 7 21 21 21 148718024 148718025 12195
32 1 1 1 31 31 31 34668543 34668544 5888
33 2 2 2 14 14 14 80730224 80730225 8985
33 8 8 8 24 24 24 322884960 322884961 17969
33 18 18 18 30 30 30 726464208 726464209 26953
37 1 1 1 11 11 11 71284248 71284249 8443
37 4 4 4 20 20 20 285103224 285103225 16885
37 9 9 9 27 27 27 641456928 641456929 25327
37 16 16 16 32 32 32 1140345360 1140345361 33769
37 25 25 25 35 35 35 1781768520 1781768521 42211
41 10 10 10 30 30 30 1187560520 1187560521 34461
45 11 11 11 33 33 33 2075986968 2075986969 45563
46 5 5 5 25 25 25 1052742915 1052742916 32446
46 20 20 20 40 40 40 4210841880 4210841881 64891
49 3 3 3 21 21 21 865124568 865124569 29413
49 12 12 12 36 36 36 3460380624 3460380625 58825
49 27 27 27 45 45 45 7785768168 7785768169 88237
50 1 1 1 13 13 13 318908163 318908164 17858
50 4 4 4 24 24 24 1275561224 1275561225 35715
50 9 9 9 33 33 33 2869959183 2869959184 53572
50 16 16 16 40 40 40 5102102040 5102102041 71429
50 25 25 25 45 45 45 7971989795 7971989796 89286
50 36 36 36 48 48 48 11479622448 11479622449 107143
51 2 2 2 18 18 18 703893960 703893961 26531
51 8 8 8 32 32 32 2815469720 2815469721 53061
51 18 18 18 42 42 42 6334727280 6334727281 79591
51 32 32 32 48 48 48 11261666640 11261666641 106121
53 13 13 13 39 39 39 5541164720 5541164721 74439
DEFDBL A-Z
CLS
FOR bse = 2 TO 53
xPart = bse * bse * bse * (bse * bse + bse + 1)
yPart = bse * bse + bse + 1
FOR x = 1 TO bse - 1
FOR y = 0 TO bse - 1
IF y <> x THEN
xxxyyy = x * xPart + y * yPart
sq = xxxyyy + 1
sr = INT(SQR(sq) + .5)
IF sr * sr = sq THEN
PRINT bse, x; x; x; y; y; y,
PRINT USING "########### ########### #######"; xxxyyy; sq; sr
END IF
END IF
NEXT
NEXT
NEXT
Edited on April 26, 2009, 5:44 pm
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Posted by Charlie
on 2009-04-26 17:42:04 |
solution and computer exploration
