## Wednesday, September 4, 2013

### Properties of A^x + B^y = C^z with Jonathan Neal

During our study of the Beal Conjecture with my friend and fellow mathematics graduate Jonathan Beal, we looked at the equation

A^x + B^y = C^z

where A, B, C, x, y, and z are integers. As a result A^x, B^y, and C^z are integers.

Let A^x be even (where A is a multiple of 2). Let B = p*m where p is a prime number. Then B^y = p^y * m^y.

The only even prime is 2. All other prime numbers (3, 5, 7, etc. ) are odd.

So if A^x is even and
p = 2 and m is even: p^y is even, m^y is even, B^y is even, and C^z is even.
p = 2 and m is odd: p^y is even, m^y is odd, B^y is even, and C^z is even.
p ≠ 2 and m is even: p^y is odd, m^y is even, B^y is even, and C^z is even.
p ≠ 2 and m is odd: p^y is odd, m^y is odd, B^y is odd, and C^z is odd

Assuming A^x is odd and
p = 2 and m is even: p^y is even, m^y is even, B^y is even, and C^z is odd.
p = 2 and m is odd: p^y is even, m^y is odd, B^y is even, and C^z is odd.
p ≠ 2 and m is even: p^y is odd, m^y is even, B^y is even, and C^z is odd.
p ≠ 2 and m is odd: p^y is odd, m^y is odd, B^y is odd, and C^z is even.

Eddie

This blog is property of Edward Shore. 2013

#### 1 comment:

1. Thank you for your post. This is excellent information. It is amazing and wonderful to visit your site. It really gives me an insight on this topic. You can find more information about residential properties here.

### TI-Nspire: Templates to Plot Functions, Parametric Equations, and Sequences

TI-Nspire:  Templates to Plot Functions, Parametric Equations, and Sequences Introduction The following are sample templates to plot functi...