a^n + b^n + c^n = x^n + y^n + z^n, n = 1,2 — (Part 2)

 

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

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

where   a,   b,   c,   x,   y,   z   are six distinct positive integers.

 

13 + 31 + 37 = 17 + 23 + 41 = 9^2 ...................... ( \,= (3\times 3)^2 \,)

13^2 + 31^2 + 37^2 = 17^2 + 23^2 + 41^2 = 2499

34 + 52 + 58 = 38 + 44 + 62 = 12^2 ..................... ( \,= (3\times 4)^2 \,)

34^2 + 52^2 + 58^2 = 38^2 + 44^2 + 62^2 = 7224

61 + 79 + 85 = 65 + 71 + 89 = 15^2 ..................... ( \,= (3\times 5)^2 \,)

61^2 + 79^2 + 85^2 = 65^2 + 71^2 + 89^2 = 17187

94 + 112 + 118 = 98 + 104 + 122 = 18^2 ................. (= (3\times 6)^2 \,)

94^2 + 112^2 + 118^2 = 98^2 + 104^2 + 122^2 = 35304

133 + 151 + 157 = 137 + 143 + 161 = 21^2 ............... (= (3\times 7)^2 \,)

133^2 + 151^2 + 157^2 = 137^2 + 143^2 + 161^2 = 65139

178 + 196 + 202 = 182 + 188 + 206 = 24^2 ............... (= (3\times 8)^2 \,)

178^2 + 196^2 + 202^2 = 182^2 + 188^2 + 206^2 = 110904

229 + 247 + 253 = 233 + 239 + 257 = 27^2 ............... (= (3\times 9)^2 \,)

229^2 + 247^2 + 253^2 = 233^2 + 239^2 + 257^2 = 177459

286 + 304 + 310 = 290 + 296 + 314 = 30^2 ............... (= (3\times 10)^2 \,)

286^2 + 304^2 + 310^2 = 290^2 + 296^2 + 314^2 = 270312

349 + 367 + 373 = 353 + 359 + 377 = 33^2 ............... (= (3\times 11)^2 \,)

349^2 + 367^2 + 373^2 = 353^2 + 359^2 + 377^2 = 395619

418 + 436 + 442 = 422 + 428 + 446 = 36^2 ............... (= (3\times 12)^2 \,)

418^2 + 436^2 + 442^2 = 422^2 + 428^2 + 446^2 = 560184

493 + 511 + 517 = 497 + 503 + 521 = 39^2 ............... (= (3\times 13)^2 \,)

493^2 + 511^2 + 517^2 = 497^2 + 503^2 + 521^2 = 771459

574 + 592 + 598 = 578 + 584 + 602 = 42^2 ............... (= (3\times 14)^2 \,)

574^2 + 592^2 + 598^2 = 578^2 + 584^2 + 602^2 = 1037544

661 + 679 + 685 = 665 + 671 + 689 = 45^2 ............... (= (3\times 15)^2 \,)

661^2 + 679^2 + 685^2 = 665^2 + 671^2 + 689^2 = 1367187

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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