x^3+ y^3 + z^3 = n^2

The goal is to find solutions of x^3+ y^3 + z^3 = n^2 where x, y, z, and n are integers greater than 1.

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

---

Define test32(k,n)=
Prgm
:x2:={}
:y2:={}
:z2:={}
:n2:={}
:For a,k,n
:For b,k,n
:For c,k,n
:d:=(a^(3)+b^(3)+c^(3))^(1/2)
:If fPart(d)=0 Then
:x2:=augment(x2,{a})
:y2:=augment(y2,{b})
:z2:=augment(z2,{c})
:n2:=augment(n2,{d})
:EndIf
:EndFor
:EndFor
:EndFor
:EndPrgm

---

Some solutions for testing integers from 1 to 50:
1^3 + 2^3 + 6^3 = 225 = 15^2
1^3 + 4^3 + 14^3 = 2809 = 53^2
2^3 + 12^3 + 50^3 = 126,736 = 356^2
3^3 + 3^3 + 3^3 = 81 = 9^2
4^3 + 4^3 + 17^3 = 5041 = 71^2
4^3 + 17^3 + 22^3 = 15,625 = 125^2
8^3 + 4^3 + 12^3 = 2304 = 48^2
8^3 + 13^3 + 15^3 = 6084 = 78^2
8^3 + 30^3 + 49^3 = 145,161 = 381^2
10^3 + 10^3 + 20^3 = 10,000 = 100^2
10^3 + 24^3 + 26^3 = 32,400 = 180^2
12^3 + 12^3 + 12^3 = 5184 = 72^2

In the range of 1 ≤ x ≤ 25, 1 ≤ y ≤ 25, and 1 ≤ z ≤ 25, there are 122 solutions.

In the range of 1 ≤ x ≤ 50, 1 ≤ y ≤ 50, and 1 ≤ z ≤ 50, there are 415 solutions. Presented next is a graph of √(x^3 + y^3 + z^3) and where square root of the sums stack up:

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

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

Plots of x^t + y^t = n^t (x, y, n are integers)

Happy Friday almost July 2013!

This is a series of plots of the following:

1. x^2 + y^2 = n^2
2. x^2 + y^2 = n^3
3. x^3 + y^3 = n^2
4. x^3 + y^2 = n^2
5. x^3 + y^2 = n^3

For x, y, and n all integers. Plots are pointed for combinations where the result, n, is an integer. The range tested is 0 ≤ x ≤ 200 and 0 ≤ y ≤ 200. I used the TI nSpire CX CAS iOS app.

How many points?

1. x^2 + y^2 = n^2, 703 points
2. x^2 + y^2 = n^3, 55 points
3. x^3 + y^3 = n^2, 97 points
4. x^3 + y^2 = n^2, 375 points
5. x^3 + y^2 = n^3, 213 points, mostly when y=0

Program Outline: (for plot 1, similar format for the rest)
Define test()
Prgm
xs:={ }
ys:={ }
For k,0,200
For l,0,200
If iPart(√(k^2+l^2))=0 Then
xs:=augment(xs,{k})
ys:=augment(ys,{l})
EndIf
EndFor
EndFor
EndPrgm


And now to the fun part... Enjoy!

That's it for now - as always thanks for your support and words of encouragement and your questions. :)

Eddie


This blog is property of Edward Shore. 2013

Happy π Day! And some Mathematical Art

Happy π Day!

π = 3.14159 26535 89793 23846 26433 83279 50288...

Who would have known that the ratio of a circle's circumference to its diameter would generate interest though out human history? π never ends, and over 10 trillion digits have been calculate. I have memorized about 15 digits. The digit zero (0) does not appear until the 32nd digit.

Why is π such a "celebrity" number? Find out here:
http://www.youtube.com/watch?v=yJ-HwrOpIps

According to Dr. James Grime, we can calculate the size of the observable universe using π with only 39 digits.
http://www.youtube.com/watch?v=FpyrF_Ci2TQ

You can use the (very slow converging series) to get π:

π = 4 × (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17...)

There are many songs about π. My favorite is "Pi" by Kate Bush (off the album "Aerial").

Tonight I plan to join desmos.com to participate in a π Day discussion.

To celebrate my birthday and π Day, I am going to share some mathematical artwork, which was created using the TI nSpire CX CAS iPad App.