Wednesday, July 10, 2013

x^2 + y^2 + z^2 = n^2

The goal is to find solutions of x^2 + y^2 + z^2 = n^2 where x, y, z, and n are integers greater than 1.

Using the following program to find solutions (TI nSpire App):



Define test22(n)=
Prgm
:x1:={}
:y1:={}
:z1:={}
:n1:={}
:For a,1,n
:For b,1,n
:For c,1,n
:d:=√(a^(2)+b^(2)+c^(2))
:If fPart(d)=0 Then
: x1:=augment(x1,{a})
: y1:=augment(y1,{b})
: z1:=augment(z1,{c})
: n1:=augment(n1,{d})
:EndIf
:EndFor
:EndFor
:EndFor
:EndPrgm


Some solutions for testing integers from 1 to 50:
1^2 + 8^2 + 32^2 = 1089 = 33^2
2^2 + 2^2 + 1^2 = 9 = 3^2
2^2 + 12^2 + 36^2 = 1444 = 38^2
3^2 + 6^2 + 6^2 = 81 = 9^2
4^2 + 7^2 + 32^2 = 1089 = 33^2
4^2 + 22^2 + 20^2 = 900 = 30^2
5^2 + 40^2 + 20^2 = 2025 = 45^2
6^2 + 2^2 + 9^2 = 121 = 11^2
6^2 + 10^2 + 15^2 = 361 = 19^2
6^2 + 42^2 + 7^2 = 1849 = 43^2
7^2 + 30^2 + 30^2 = 1849 = 43^2
8^2 + 5^2 + 44^2 = 2025 = 45^2
8^2 + 24^2 + 27^2 = 1369 = 37^2
9^2 + 30^2 + 50^2 = 3481 = 59^2

In the range of 1 ≤ x ≤ 25, 1 ≤ y ≤ 25, and 1 ≤ z ≤ 25, there are 288 solutions.

In the range of 1 ≤ x ≤ 50, 1 ≤ y ≤ 50, and 1 ≤ z ≤ 50, there are 1209 solutions. Presented next is a graph of √(x^2 + y^2 + z^2) and where square root of the sums stack up:

Note that are two results (n) that show up the most.

For the range of 1 to 25, the dominating n's are 21 and 23, each resulting showing 33 times in the 288 solutions.

However, the most dominated results shifts as the range increases. For the range of 1 to 50, the dominating results are 45 and 51, each showing up 60 times in the 1209 solutions.

Finally, using the MathStudio app, is a plot of solutions for 0 ≤ x ≤ 25, 0 ≤ y ≤ 25, and 0 ≤ z ≤ 25.

Until next time,

Eddie


This blog is property of Edward Shore. 2013

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