Integers: x^5 + y^5 = n^2, x^2 + y^2 = n^5, x^2 + y^3 = n^2 and x^3 + y^2 = n^3

Let x, y, and n be integers.

x^2 + y^2 = n^5

x and y tested: 1 ≤ x ≤ 30, 1 ≤ y ≤ 30

What I found: (there may be more outside the range I tested)
4^2 + 4^2 = 2^5
(Thank you Bill Zimmerly for pointing out my typo! (32 is not 5^3))

HP 32sii Program:

LBL E
x^2
x<>y
x^2
+
5
x√y
RTN

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x^5 + y^5 = n^2

x and y tested: 1 ≤ x ≤ 30, 1 ≤ y ≤ 30

Again, I am looking for combinations where x, y, and n are all integers.

In the range tested I found:
2^5 + 2^5 = 8^2
8^5 + 8^5 = 256^2
18^5 + 18^5 = 1944^2

HP 32Sii Program:

LBL F
5
y^x
x<>y
5
y^x
+
√
RTN

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The next program will cover two explorations. I will present the program first, then the equations.

HP32Sii Program:
LBL G
x^2
x<>y
3
y^x
+
ENTER
√ \\ √(x^2 + y^3)
R/S
x<>y
3
x√y \\cube root of (x^2 + y^3)
RTN

I used this program for two tests. The area tested was 1 ≤ x ≤ 16 and 1 ≤ y ≤ 16. I am looking for combinations where x, y, and n are integers.

x^2 + y^3 = n^2

2^3 + 1^2 = 3^2
3^3 + 3^2 = 6^2
4^3 + 6^2 = 10^2
5^3 + 10^2 = 15^2
6^3 + 3^2 = 15^2
8^3 + 8^2 = 24^2
12^3 + 6^2 = 42^2
12^3 + 11^2 = 43^2
3^3 + 13^2 = 14^2
4^3 + 15^2 = 17^2
6^3 + 15^2 = 21^2
10^3 + 15^2 = 35^2
15^3 + 15^2 = 60^2

x^2 + y^3 = n^3

7^3 + 13^2 = 60^2

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Happy June Solstice!

Eddie


This blog is property of Edward Shore. 2013