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
A blog is that is all about mathematics and calculators, two of my passions in life.
Wednesday, September 4, 2013
Properties of A^x + B^y = C^z with Jonathan Neal
Subscribe to:
Post Comments (Atom)
Casio fx3650P: Circular Segment
Casio fx3650P: Circular Segment Introduction Variables: X: radius Y: angle (in degree) C: chord length D: altitude A:...

Casio fx991EX Classwiz Review Casio FX991EX The next incarnation of the fx991 line of Casio calculators is the fx991 EX. ...

The Odds of Hitting it Big The number of possible combinations is fairly easy to calculate. You multiply the number symbols each slot has ...

HP Prime: Basic CAS Commands for Polynomials and Rational Expressions Define the following variables: poly: a polynomial o...
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.
ReplyDelete