Little math problems September 2018

[Valentin Albillo]

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Hi, A.Chan:

(09-24-2018 12:18 PM)Albert Chan Wrote:  

(09-24-2018 12:53 AM)Valentin Albillo Wrote:        (2,2), (128, 128), (1250, 625), (1458, 1458), (8192, 8192), (31250, 31250), (80000, 40000)

and infinitely more can be found by simply expanding the search range, though at this point it's more efficient (albeit more convoluted) to begin using congruences and parameterization, as stated.

Is there a pattern to the solution pairs ? For above, it seems to be (2 N^3, 2 N^3) or (2 N^3, N^3).

There's an infinity of solutions that do not fit that particular pattern, for instance ( 3430000, 10290000 ):

      3430000 ^2 + 10290000 ^2 = 117649000000000 = 49000 ^3
      3430000 ^3 + 10290000 ^3 = 1129900996000000000000 = 33614000000 ^2

Quote:What divisibility criteria did you end up using ?

It depends on the particular case being analyzed.

Regards.
V.
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