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

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

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

Happy June Solstice!

Eddie

This blog is property of Edward Shore. 2013