Make {a^2 + b, a + b^2} squares

 
 
Find two distinct rational numbers   (a, \; b)   to make the two of the expressions squares

                                                 a^2 \; + \; b
                                                 a \; + \; b^2

 

Here are some solutions

x^2+y 1
x^2+y 2

 
 

Steve Kass found a pair   (144/299,  155/299)   where   M = N

(144/299)^2 \; + \; (155/299) \; = \; (144/299) \; + \; (155/299)^2 \; = \; (259/299)^2

 

I found another pair :   (3/8,   5/8)

(3/8)^2 \; + \; (5/8) \; = \; (3/8) \; + \; (5/8)^2 \; = \; (7/8)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

3 Responses to Make {a^2 + b, a + b^2} squares

  1. Paul says:

    Here are a few, first set bottom heavy, last set top heavy

    2/25^2 + 17/100 = 21/50^2 & 2/25 + 17/100^2 = 33/100^2
    3/25^2 + 13/100 = 19/50^2 & 3/25 + 13/100^2 = 37/100^2
    3/64^2 + 13/64 = 29/64^2 & 3/64 + 13/64^2 = 19/64^2
    4/25^2 + 9/100 = 17/50^2 & 4/25 + 9/100^2 = 41/100^2
    5/64^2 + 11/64 = 27/64^2 & 5/64 + 11/64^2 = 21/64^2
    7/64^2 + 9/64 = 25/64^2 & 7/64 + 9/64^2 = 23/64^2

    5/4^2 + 19/9 = 23/12^2 & 5/4 + 19/9^2 = 43/18^2
    14/9^2 + 65/36 = 37/18^2 & 14/9 + 65/36^2 = 79/36^2
    16/9^2 + 16/9 = 20/9^2 & 16/9 + 16/9^2 = 20/9^2
    19/9^2 + 5/4 = 43/18^2 & 19/9 + 5/4^2 = 23/12^2
    21/16^2 + 25/16 = 29/16^2 & 21/16 + 25/16^2 = 31/16^2
    25/4^2 + 100/9 = 85/12^2 & 25/4 + 100/9^2 = 205/18^2
    25/16^2 + 21/16 = 31/16^2 & 25/16 + 21/16^2 = 29/16^2
    25/16^2 + 99/16 = 47/16^2 & 25/16 + 99/16^2 = 101/16^2
    40/9^2 + 89/9 = 49/9^2 & 40/9 + 89/9^2 = 91/9^2
    65/36^2 + 14/9 = 79/36^2 & 65/36 + 14/9^2 = 37/18^2
    89/9^2 + 40/9 = 91/9^2 & 89/9 + 40/9^2 = 49/9^2
    99/16^2 + 25/16 = 101/16^2 & 99/16 + 25/16^2 = 47/16^2
    100/9^2 + 25/4 = 205/18^2 & 100/9 + 25/4^2 = 85/12^2

    Paul.

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