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Square products (Posted on 2018-02-27) Difficulty: 3 of 5
Find all integer solutions to (x3-1)(y3-1)=z2, with |x|<|y| and z>0.

No Solution Yet Submitted by Math Man    
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re: computer exploration / thoughts | Comment 2 of 3 |
(In reply to computer exploration by Charlie)

There doesn't seem to be much to go on here.  The solutions look like coincidences rather than a pattern.


The last two coming from
(3^3-1)=2*13
(313^3-1)=2^3*3^2*13*181^2
filling just the right gaps so the product is
(2^2*3*13*181)^2

and
(20^3+1)=3^2*7*127
(362^3+1)=3^2*7^3*11^2*127
product
(3^2*7^2*11*127)^2

Other choices for x leave factors that may or may not ever be filled by some y.  For example
(11^3-1)=2*5*7*19
Does some large y give a number have these prime factors to odd powers and any others to even powers?
Using y=2*5*7*19+1 gives each factor (difference of cubes) but doesn't guarantee powers.


  Posted by Jer on 2018-02-28 15:19:12
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