Can you find 3 distinct rational numbers to can make the three expressions squares

Can you find 3 distinct rational numbers to can make the three expressions squares

Let **P** be the perimeter

**(1)**

Can you explain the following:

The generators and of the first few triangles are the Pell numbers

2, 5, 12, 29, …, 13860, and 1, 2, 5, …, 5741, respectively.

The lengths of their hypotenuses are the Pell numbers, 5, 29, 169, …

A000129

*1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149, 107578520350, 259717522849, … *

Find three positive integers to can make the two expressions squares

Here’s a parametric solution established by Euler:

In two ways:

Find three positive integers to can make the three expressions squares

Can you find three **distinct** positive integers that satisfy the system of equations?

we can make the three expressions squares

(1)

(2)

(3)

with a simple parameterization:

(1)

(2)

(3)

Interestingly,

(2) + (1)

(3)

Find three distinct positive integers to make the three expressions squares

Here are some solutions

Find other solutions

are positive integers

a simple parameterization

Find two distinct rational numbers to make the two of the expressions squares

Here are some solutions

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

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

Find positive integers such that

are made to be squares

for

(a, b, c, d) = (**3**, 14, 8, 5)

(a, b, c, d) = (**5**, 38, 21, 16)

(a, b, c, d) = (**8**, 14, 5, 3)

(a, b, c, d) = (**8**, 26, 15, 7)

(a, b, c, d) = (**11**, 62, 35, 24)

(a, b, c, d) = (**13**, 86, 48, 35)

(a, b, c, d) = (**15**, 26, 7, 8)

(a, b, c, d) = (**15**, 26, 8, 7)

(a, b, c, d) = (**16**, 38, 21, 5)

(a, b, c, d) = (**21**, 38, 5, 16)

(a, b, c, d) = (**24**, 62, 35, 11)

(a, b, c, d) = (**32**, 134, 77, 45)

(a, b, c, d) = (**33**, 74, 40, 7)

(a, b, c, d) = (**35**, 62, 11, 24)

(a, b, c, d) = (**39**, 98, 55, 16)

(a, b, c, d) = (**40**, 74, 7, 33)

(a, b, c, d) = (**40**, 74, 33, 7)

(a, b, c, d) = (**48**, 86, 13, 35)

(a, b, c, d) = (**48**, 86, 35, 13)

(a, b, c, d) = (**55**, 98, 39, 16)

(a, b, c, d) = (**56**, 122, 65, 9)

(a, b, c, d) = (**65**, 122, 56, 9)

(a, b, c, d) = (**77**, 134, 32, 45)

(a, b, c, d) = (**80**, 146, 17, 63)

(a, b, c, d) = (**80**, 146, 63, 17)

(a, b, c, d) = (**80**, 182, 99, 19)

(a, b, c, d) = (**88**, 266, 153, 65)

(a, b, c, d) = (**91**, 158, 51, 40)

(a, b, c, d) = (**96**, 182, 11, 85)

(a, b, c, d) = (**96**, 182, 85, 11)

(a, b, c, d) = (**99**, 182, 19, 80)

Find other solutions.