## 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

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)

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

math grad - Interest: Number theory
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### 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.

• benvitalis says:

Nice work. Steve found an interesting result.

• benvitalis says:

I found another pair: (3/8, 5/8)