a^n + b^n + c^n = x^n + y^n + z^n, where n = 2, 3

 

Find six distinct positive integers   a, \; b, \; c, \; x, \; y, \; z   such that

       a^2 \; + \; b^2 \; + \; c^2 \; = \; x^2 \; + \; y^2 \; + \; z^2
       a^3 \; + \; b^3 \; + \; c^3 \; = \; x^3 \; + \; y^3 \; + \; z^3

 
 

Note that in the following examples   d \; = \; (x+y+z) \; - \; (a+b+c) \; = \; 12

1^2 \; + \; 54^2 \; + \; 69^2 \; = \; 18^2 \; + \; 45^2 \; + \; 73^2 \; = \; 7678
1^3 \; + \; 54^3 \; + \; 69^3 \; = \; 18^3 \; + \; 45^3 \; + \; 73^3 \; = \; 485974

1 + 54 + 69 = 124 ……. 18 + 45 + 73 = 136 ……. d = 136 – 124 = 12
 

2^2 \; + \; 45^2 \; + \; 62^2 \; = \; 26^2 \; + \; 29^2 \; + \; 66^2 \; = \; 5873
2^3 \; + \; 45^3 \; + \; 62^3 \; = \; 26^3 \; + \; 29^3 \; + \; 66^3 \; = \; 329461

2 + 45 + 62 = 109 ……. 26 + 29 + 66 = 121 ……. d = 121 – 109 = 12
 

3^2 \; + \; 58^2 \; + \; 69^2 \; = \; 22^2 \; + \; 45^2 \; + \; 75^2 \; = \; 8134
3^3 \; + \; 58^3 \; + \; 69^3 \; = \; 22^3 \; + \; 45^3 \; + \; 75^3 \; = \; 523648

3 + 58 + 69 = 130 ……. 22 + 45 + 75 = 142 ……. d = 142 – 130 = 12
 

4^2 \; + \; 59^2 \; + \; 81^2 \; = \; 23^2 \; + \; 48^2 \; + \; 85^2 \; = \; 10058
4^3 \; + \; 59^3 \; + \; 81^3 \; = \; 23^3 \; + \; 48^3 \; + \; 85^3 \; = \; 736884

4 + 59 + 81 = 144 ……. 23 + 48 + 85 = 156 ……. d = 156 – 144 = 12
 

6^2 \; + \; 51^2 \; + \; 97^2 \; = \; 33^2 \; + \; 34^2 \; + \; 99^2 \; = \; 12046
6^3 \; + \; 51^3 \; + \; 97^3 \; = \; 33^3 \; + \; 34^3 \; + \; 99^3 \; = \; 1045540

6 + 51 + 97 = 154 ……. 33 + 34 + 99 = 166 ……. d = 166 – 154 = 12
 

9^2 \; + \; 69^2 \; + \; 86^2 \; = \; 33^2 \; + \; 50^2 \; + \; 93^2 \; = \; 12238
9^3 \; + \; 69^3 \; + \; 86^3 \; = \; 33^3 \; + \; 50^3 \; + \; 93^3 \; = \; 965294

9 + 69 + 86 = 164 ……. 33 + 50 + 93 = 176 ……. d = 176 – 164 = 12

 

Find other six distinct positive integers that satisfy the two equations where   d = 12

How about when   d   is a multiple of 12?

Is it possible to find such integers when   d   is not a multiple of 12?

 
 

Paul found:

10^2 \; + \; 67^2 \; + \; 102^2 \; = \; 34^2 \; + \; 51^2 \; + \; 106^2 \; = \; 14993
10^3 \; + \; 67^3 \; + \; 102^3 \; = \; 34^3 \; + \; 51^3 \; + \; 106^3 \; = \; 1362971

10 + 67 + 102 = 179 …….. 34 + 51 + 106 = 191 …….. d = 191 – 179 = 12

15^2 \; + \; 100^2 \; + \; 118^2 \; = \; 36^2 \; + \; 82^2 \; + \; 127^2 \; = \; 24149
15^3 \; + \; 100^3 \; + \; 118^3 \; = \; 36^3 \; + \; 82^3 \; + \; 127^3 \; = \; 2646407

15 + 100 + 118 = 233 ……. 36 + 82 + 127 = 245 …….. d = 245 – 233 = 12

23^2 \; + \; 114^2 \; + \; 118^2 \; = \; 50^2 \; + \; 82^2 \; + \; 135^2 \; = \; 27449
23^3 \; + \; 114^3 \; + \; 118^3 \; = \; 50^3 \; + \; 82^3 \; + \; 135^3 \; = \; 3136743

23 + 114 + 118 = 255 ……. 50 + 82 + 135 = 267 …….. d = 267 – 255 = 12

29^2 \; + \; 108^2 \; + \; 143^2 \; = \; 65^2 \; + \; 75^2 \; + \; 152^2 \; = \; 32954
29^3 \; + \; 108^3 \; + \; 143^3 \; = \; 65^3 \; + \; 75^3 \; + \; 152^3 \; = \; 4208308

29 + 108 + 143 = 280 ……. 65 + 75 + 152 = 292 …….. d = 292 – 280 = 12

8^2 \; + \; 78^2 \; + \; 169^2 \; = \; 25^2 \; + \; 72^2 \; + \; 170^2 \; = \; 34709
8^3 \; + \; 78^3 \; + \; 169^3 \; = \; 25^3 \; + \; 72^3 \; + \; 170^3 \; = \; 5301873

8 + 78 + 169 = 255 ……… 25 + 72 + 170 = 267 …….. d = 267 – 255 = 12

27^2 \; + \; 135^2 \; + \; 142^2 \; = \; 51^2 \; + \; 106^2 \; + \; 159^2 \; = \; 39118
27^3 \; + \; 135^3 \; + \; 142^3 \; = \; 51^3 \; + \; 106^3 \; + \; 159^3 \; = \; 5343346

27 + 135 + 142 = 304 ……. 51 + 106 + 159 = 316 ……. d = 316 – 304 = 12

51^2 \; + \; 166^2 \; + \; 167^2 \; = \; 99^2 \; + \; 103^2 \; + \; 194^2 \; = \; 58046
51^3 \; + \; 166^3 \; + \; 167^3 \; = \; 99^3 \; + \; 103^3 \; + \; 194^3 \; = \; 9364410

51 + 166 + 167 = 384 ……. 99 + 103 + 194 = 396 ……. d = 396 – 384 = 12

44^2 \; + \; 153^2 \; + \; 183^2 \; = \; 75^2 \; + \; 120^2 \; + \; 197^2 \; = \; 58834
44^3 \; + \; 153^3 \; + \; 183^3 \; = \; 75^3 \; + \; 120^3 \; + \; 197^3 \; = \; 9795248

44 + 153 + 183 = 380 ……. 75 + 120 + 197 = 392 ……. d = 392 – 380 = 12

 

                                                             d \; = \; 24

 

4^2 \; + \; 90^2 \; + \; 124^2 \; = \; 52^2 \; + \; 58^2 \; + \; 132^2 \; = \; 23492
4^3 \; + \; 90^3 \; + \; 124^3 \; = \; 52^3 \; + \; 58^3 \; + \; 132^3 \; = \; 2635688

4 + 90 + 124 = 218 …….. 52 + 58 + 132 = 242 …….. d = 242 – 218 = 24

2^2 \; + \; 108^2 \; + \; 138^2 \; = \; 36^2 \; + \; 90^2 \; + \; 146^2 \; = \; 30712
2^3 \; + \; 108^3 \; + \; 138^3 \; = \; 36^3 \; + \; 90^3 \; + \; 146^3 \; = \; 3887792

2 + 108 + 138 = 248 ……. 36 + 90 + 146 = 272 …….. d = 272 – 248 = 24

6^2 \; + \; 116^2 \; + \; 138^2 \; = \; 44^2 \; + \; 90^2 \; + \; 150^2 \; = \; 32536
6^3 \; + \; 116^3 \; + \; 138^3 \; = \; 44^3 \; + \; 90^3 \; + \; 150^3 \; = \; 4189184

6 + 116 + 138 = 260 ……. 44 + 90 + 150 = 284 …….. d = 284 – 260 = 24

8^2 \; + \; 118^2 \; + \; 162^2 \; = \; 46^2 \; + \; 96^2 \; + \; 170^2 \; = \; 40232
8^3 \; + \; 118^3 \; + \; 162^3 \; = \; 46^3 \; + \; 96^3 \; + \; 170^3 \; = \; 5895072

8 + 118 + 162 = 288 ……. 46 + 96 + 170 = 312 …….. d = 312 – 288 = 24

4^2 \; + \; 124^2 \; + \; 165^2 \; = \; 37^2 \; + \; 108^2 \; + \; 172^2 \; = \; 42617
4^3 \; + \; 124^3 \; + \; 165^3 \; = \; 37^3 \; + \; 108^3 \; + \; 172^3 \; = \; 6398813

4 + 124 + 165 = 293 ……. 37 + 108 + 172 = 317 ……. d = 317 – 293 = 24

12^2 \; + \; 102^2 \; + \; 194^2 \; = \; 66^2 \; + \; 68^2 \; + \; 198^2 \; = \; 48184
12^3 \; + \; 102^3 \; + \; 194^3 \; = \; 66^3 \; + \; 68^3 \; + \; 198^3 \; = \; 8364320

12 + 102 + 194 = 308 …… 66 + 68 + 198 = 332 …….. d = 332 – 308 = 24

18^2 \; + \; 138^2 \; + \; 172^2 \; = \; 66^2 \; + \; 100^2 \; + \; 186^2 \; = \; 48952
18^3 \; + \; 138^3 \; + \; 172^3 \; = \; 66^3 \; + \; 100^3 \; + \; 186^3 \; = \; 7722352

18 + 138 + 172 = 328 …… 66 + 100 + 186 = 352 ……. d = 352 – 328 = 24

12^2 \; + \; 132^2 \; + \; 179^2 \; = \; 51^2 \; + \; 108^2 \; + \; 188^2 \; = \; 49609
12^3 \; + \; 132^3 \; + \; 179^3 \; = \; 51^3 \; + \; 108^3 \; + \; 188^3 \; = \; 8037035

12 + 132 + 179 = 323 …… 51 + 108 + 188 = 347 ……. d = 347 – 323 = 24

24^2 \; + \; 147^2 \; + \; 182^2 \; = \; 83^2 \; + \; 96^2 \; + \; 198^2 \; = \; 55309
24^3 \; + \; 147^3 \; + \; 182^3 \; = \; 83^3 \; + \; 96^3 \; + \; 198^3 \; = \; 9218915

24 + 147 + 182 = 353 …… 83 + 96 + 198 = 377 …….. d = 377 – 353 = 24

 

 

                                                             d \; = \; 36

 

6^2 \; + \; 135^2 \; + \; 186^2 \; = \; 78^2 \; + \; 87^2 \; + \; 198^2 \; = \; 52857
6^3 \; + \; 135^3 \; + \; 186^3 \; = \; 78^3 \; + \; 87^3 \; + \; 198^3 \; = \; 8895447

6 + 135 + 186 = 327 ……. 78 + 87 + 198 = 363 ……. d = 363 – 327 = 36

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

7 Responses to a^n + b^n + c^n = x^n + y^n + z^n, where n = 2, 3

  1. Paul says:

    Here’s a few more. Format is {{a, b, c}, {x, y, z}}

    {{10,67,102},{34,51,106}}, d = 12
    {{4,90,124},{52,58,132}}, d = 24
    {{15,100,118},{36,82,127}}, d = 12
    {{23,114,118},{50,82,135}}, d = 12
    {{2,108,138},{36,90,146}}, d = 24
    {{6,116,138},{44,90,150}}, d = 24
    {{29,108,143},{65,75,152}}, d = 12
    {{8,78,169},{25,72,170}}, d = 12
    {{27,135,142},{51,106,159}}, d = 12
    {{8,118,162},{46,96,170}}, d = 24
    {{4,124,165},{37,108,172}}, d = 24
    {{12,102,194},{66,68,198}}, d = 24
    {{18,138,172},{66,100,186}}, d = 24
    {{12,132,179},{51,108,188}}, d = 24
    {{6,135,186},{78,87,198}}, d = 36
    {{24,147,182},{83,96,198}}, d = 24
    {{51,166,167},{99,103,194}}, d = 12
    {{44,153,183},{75,120,197}}, d = 12

    Seems unlikely that d will be other than a multiple of 12.

    Paul.

  2. Paul says:

    I have extended the range from the last entry, it is in an excel sheet in dropbox, called “a to the n”
    here is a link too. I have extended the sum of squares to 500000 and all differences are still multiples of 12. with differences up to 120.

    https://www.dropbox.com/s/c42jkairww6e4m5/a%20to%20the%20n.xlsx?dl=0

    Paul.

    • David @InfinitelyManic says:

      Paul – how long did your code take to complete the dropbox list?

    • David @InfinitelyManic says:

      Thanks – I was able to run through five nested loops from 1 to 200 in about 6 hours on a Cortex-A9 non-virtual server using ARMv7 Assembly. I see you reached higher numbers but I was just curious about host much more optimization could have been achieved.

  3. David @InfinitelyManic says:

    There may be few new gems in there:
    x y z a b c

    2 45 62 26 29 66
    2 108 138 36 90 146
    2 220 267 102 155 292
    2 225 241 110 141 277
    3 58 69 22 45 75
    3 162 207 54 135 219
    3 275 398 83 243 410
    4 59 81 23 48 85
    4 90 124 52 58 132
    4 124 165 37 108 172
    4 164 393 76 137 396
    4 216 276 72 180 292
    ..

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