Problem: Squares Roots Relation (click here to see the problem statement)

My Solution.

I rewrote the equality to remove some of the radicals:

 

Then, using the "simplified" statement above, I create a a spreadsheet to look for patterns. I fixed the x-values and adjusted the y-values. In the tables below, I fixed x at 1, 2, and 3:

 

x

y

z
 x

y

z
 x

y

z
1 1 4 2 2 8 3 3 12
1 2 5.82842712474619 2 3 9.89897948556636 3 4 13.9282032302755
1 3 7.46410161513775 2 4 11.6568542494924 3 5 15.7459666924148
1 4 9 2 5 13.3245553203368 3 6 17.4852813742386
1 5 10.4721359549996 2 6 14.9282032302755 3 7 19.1651513899117
1 6 11.8989794855664 2 7 16.4833147735479 3 8 20.7979589711327
1 7 13.2915026221292 2 8 18 3 9 22.3923048454133
1 8 14.6568542494924 2 9 19.4852813742386 3 10 23.9544511501033
1 9 16 2 10 20.9442719099992 3 11 25.4891252930761
1 10 17.3245553203368 2 11 22.3808315196469 3 12 27
1 11 18.6332495807108 2 12 23.7979589711327 3 13 28.4899959967968
1 12 19.9282032302755 2 13 25.1980390271856 3 14 29.9614813968157
1 13 21.211102550928 2 14 26.5830052442584 3 15 31.4164078649987
1 14 22.4833147735479 2 15 27.9544511501033 3 16 32.856406460551
1 15 23.7459666924148 2 16 29.3137084989848 3 17 34.2828568570857
1 16 25 2 17 30.6619037896906 3 18 35.6969384566991
1 17 26.2462112512353 2 18 32 3 19 37.0996688705415
1 18 27.4852813742386 2 19 33.328828005938 3 20 38.4919333848297
1 19 28.7177978870813 2 20 34.6491106406735 3 21 39.8745078663875
1 20 29.9442719099992 2 21 35.9614813968157 3 22 41.2480768092719
1 21 31.1651513899117 2 22 37.2664991614216 3 23 42.6132477258361
1 22 32.3808315196469 2 23 38.5646599662505 3 24 43.9705627484771
1 23 33.5916630466254 2 24 39.856406460551 3 25 45.3205080756888
1 24 34.7979589711327 2 25 41.142135623731 3 26 46.6635217326557
1 25 36 2 26 42.422205101856 3 27 48
1 26 37.1980390271856 2 27 43.6969384566991 3 28 49.3303027798234
1 27 38.3923048454133 2 28 44.9666295470958 3 29 50.6547581061776
1 28 39.5830052442584 2 29 46.2315462117278 3 30 51.9736659610103
1 29 40.770329614269 2 30 47.4919333848297 3 31 53.2873015219859
1 30 41.9544511501033 2 31 48.7480157480236 3 32 54.5959179422654
1 31 43.13552872566 2 32 50 3 33 55.8997487421324
1 32 44.3137084989848 2 33 51.2480768092719 3 34 57.1990098767242
1 33 45.4891252930761 2 34 52.4924225024706 3 35 58.4939015319192
1 34 46.6619037896906 2 35 53.7332005306815 3 36 59.7846096908265
1 35 47.8321595661992 2 36 54.9705627484771 3 37 61.0713075057055
1 36 49 2 37 56.2046505340853 3 38 62.3541565040626
1 37 50.1655250605964 2 38 57.4355957741627 3 39 63.6333076527839
1 38 51.328828005938 2 39 58.6635217326557 3 40 64.9089023002066
1 39 52.4899959967968 2 40 59.8885438199983 3 41 66.1810730128188
1 40 53.6491106406735 2 41 61.1107702762748 3 42 67.4499443206437
1 41 54.8062484748657 2 42 62.3303027798234 3 43 68.7156333832011
1 42 55.9614813968157 2 43 63.5472369909914 3 44 69.9782505861521
1 43 57.114877048604 2 44 64.7616630392937 3 45 71.2379000772445
1 44 58.2664991614216 2 45 65.9736659610103 3 46 72.4946802489415
1 45 59.4164078649987 2 46 67.1833260932509 3 47 73.7486841740758
1 46 60.5646599662505 2 47 68.3907194296653 3 48 75
1 47 61.7113092008021 2 48 69.5959179422654 3 49 76.2487113059643
1 48 62.856406460551 2 49 70.7989898732233 3 50 77.4948974278318
1 49 64 2 50 72 3 51 78.738633753706
1 50 65.142135623731 2 51 73.1990098767241 3 52 79.9799919935936
1 51 66.2828568570857 2 52 74.3960780543711 3 53 81.219040425837
1 52 67.422205101856 2 53 75.591260281974 3 54 82.4558441227157
1 53 68.560219778561 2 54 76.7846096908265 3 55 83.6904651573303

I soon noticed that when x=y on my initial iteration, the resulting z-value appears to consistently equal the y-value that will next give an integer value for z, given some fixed value for x:

 

So I constructed another spreadsheet using the resulting z-value as the input for the succeeding y-value:

x

y

z

x

y

z

x

y

z

x

y

z
1 1 4 2 2 8 3 3 12 4 4 16
1 4 9 2 8 18 3 12 27 4 16 36
1 9 16 2 18 32 3 27 48 4 36 64
1 16 25 2 32 50 3 48 75 4 64 100
1 25 36 2 50 72 3 75 108 4 100 144
1 36 49 2 72 98 3 108 147 4 144 196
1 49 64 2 98 128 3 147 192 4 196 256
1 64 81 2 128 162 3 192 243 4 256 324
1 81 100 2 162 200 3 243 300 4 324 400
1 100 121 2 200 242 3 300 363 4 400 484
1 121 144 2 242 288 3 363 432 4 484 576
1 144 169 2 288 338 3 432 507 4 576 676
1 169 196 2 338 392 3 507 588 4 676 784
1 196 225 2 392 450 3 588 675 4 784 900
1 225 256 2 450 512 3 675 768 4 900 1024
1 256 289 2 512 578 3 768 867 4 1024 1156

 

 

aaa

An Algebraic Representation

 

so and

so and

so and

 

 

 

The pattern of relationships between a fixed x, and successive y- and z-values is shown in the table below:

 n

xn

yn

zn

1

x

x

4x

2

x

4x

9x

3

x

9x

16x

4

x

16x

25x

5

x

25x

36x

 

We can generalize the relationship as follows:

 

 

 

 

aaa

A Graphical Representation

 

If we use the "middle" equation of my simplified equations given at the top of this discussion >

 

We can construct a graphical representation based on that equation:

=

 

IF we assign y = x on the first interation we have: = = 4x

 

 

This result for z then becomes the y-value on the next interation:

= = = = 9x

 

 

Again, this result for z becomes the y-value on the next interation:

= = = 16x

And so on.