(x^2 – 1) (y^2 – 1) = (z^2 – 1)^2

 
  
(x^2 - 1) \, (y^2 - 1) \; = \; (z^2 - 1)^2

An alternate form assuming   x, y,   and   z   are positive:

x^2 \, y^2 \; - \; x^2 \; - \; y^2 \; + \; 1 \; = \; (z^2 - 1)^2     or

x^2 \, y^2 \; + \; 1 \; = \; x^2 \; + \; y^2 \; + \; (z^2 - 1)^2

 

Here are the first few solutions:

 
(17^2 - 1) \,(3^2 - 1) = (7^2 - 1)^2
(99^2 - 1) \,(17^2 - 1) = (41^2 - 1)^2
(577^2 - 1) \,(99^2 - 1) = (239^2 - 1)^2
(3363^2 - 1) \,(577^2 - 1) = (1393^2 - 1)^2
(19601^2 - 1) \,(3363^2 - 1) = (8119^2 - 1)^2
(114243^2 - 1) \,(19601^2 - 1) = (47321^2 - 1)^2
(665857^2 - 1) \,(114243^2 - 1) = (275807^2 - 1)^2
(3880899^2 - 1) \,(665857^2 - 1) = (1607521^2 - 1)^2
(22619537^2 - 1) \,(3880899^2 - 1) = (9369319^2 - 1)^2
(131836323^2 - 1) \,(22619537^2 - 1) = (54608393^2 - 1)^2
(768398401^2 - 1) \,(131836323^2 - 1) = (318281039^2 - 1)^2
(4478554083^2 - 1) \,(768398401^2 - 1) = (1855077841^2 - 1)^2
(26102926097^2 - 1) \,(4478554083^2 - 1) = (10812186007^2 - 1)^2
(152139002499^2 - 1) \,(26102926097^2 - 1) = (63018038201^2 - 1)^2
(886731088897^2 - 1) \,(152139002499^2 - 1) = (367296043199^2 - 1)^2
(5168247530883^2 - 1) \,(886731088897^2 - 1) = (2140758220993^2 - 1)^2
(30122754096401^2 - 1) \,(5168247530883^2 - 1) = (12477253282759^2 - 1)^2
(175568277047523^2 - 1) \,(30122754096401^2 - 1) = (72722761475561^2 - 1)^2
(1023286908188737^2 - 1) \,(175568277047523^2 - 1) = (423859315570607^2 - 1)^2
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 

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About benvitalis

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

One Response to (x^2 – 1) (y^2 – 1) = (z^2 – 1)^2

  1. John McMahon says:

    The equation

    (x^2 - 1) \, (y^2 - 1) \; = \; (z^2 - 1)^2

    Is equivalent to:

    (Q_{(2n)^2} \; - \; 1) \,(Q_{(2n + 2)^2} \; - \; 1) \; = \; (Q_{(2n + 1)^2} \; - \; 1)^2

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