Beal Conjecture Case Study Proof 2015 and 2013 Beal Conjecture Disproven-Planet Number Different Based on Larger than Pluto Standard, Beal Disproven Based on Examples and 2015 Proof Based on Examples, 2015 Disproof Based on Disproof of If Statements Added

Disproving Conventional 9 Planet Concepts Our Solar System would have 15 planets using the Pluto Standard of Pluto's 2200 km diameter. Jupiter, Io, Europa, Callisto, Ganymede would be a Quintuple Planet or 5 planets, Saturn and Titan a Double Planet or 2 planets, Neptune and Triton also a Double Planet or 2 planets. By the way using Charon's 1200 km diameter as the standard for planets, our solar system has 19 planets. Standards used or applied, seeing a planet as a certain diameter or planets as having a certain diameter like Pluto's or Charon's, can change the world. Beal Conjecture can be disproved using specific cases. Disproof of Beal Conjecture At about 7:20pm May 29, 2013 in Eckhart Library University of Chicago, I became aware of a prize for disproving or proving the Beal Conjecture. I had been writing some songs like "Song Pong" and "I asked My God to Sing my Song" and inventing a new sport called “Prayer to Someone Sport.” I thought that even though I did not know where my proof was going to go that I could sit down and develop a coherent disproof of the conjecture just writing down some ideas that could disprove it.

Disproof of the Beal Conjecture From http://www.math.unt.edu/~mauldin/beal.html accessed on 5/29/2013 at 7:42 library closes at 8 pm BEAL'S CONJECTURE: BEAL'S CONJECTURE:

BEAL'S CONJECTURE: If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.

THE BEAL PRIZE. The conjecture and prize was announced in the December 1997 issue of the Notices of the American Mathematical Society. Since that time Andy Beal has increased the amount of the prize for his conjecture. The prize is now this: $100,000 for either a proof or a counterexample of his conjecture. The prize money is being held by the American Mathematical Society until it is awarded. In the meantime the interest is being used to fund some AMS activities and the annual Erdos Memorial Lecture.

Revised Disproofs and Proof 4/18/2015

BEAL'S CONJECTURE: If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.

Disproof #1 If we raise A as integer to infinity and B as integer raised to infinity and C as integer raised to infinity, then the equation does not equal as two infinities added together will not equal one infinity. Beal Conjecture thereby disproved since the initial statement Ax +By = Cz Has a case that is false. 1 infinity added to 1 infinity is not identical to 1 infinity as the first side of the equation is a larger infinity.

Disproof #2 A, B, C defined as 0 which is positive integer. Equation is true for Beal's If statement, but ABC do not have common prime number factors, 0 is not a prime although initial statement true. 0 is an integer, but not a prime number

Disproof #3 (Developed Later on May 29, 2013) A=2 , B=2, C=2 x=3,y=3, z=4 Ax +By = Cz. That is, 2 raised to the 3rd powers=8, 2 raised to the third power = 3rd power equals 8 and 2 raised to the 4th power equals 16. There is more than one common prime factor of A,B,C. The language of the conjecture is false as A,B, C has more than one prime factors of 2, 2, and 2 and 2. As there is more than one prime factor, the conjecture is disproven by there being more than one common prime factor. Beal's statement can be seen as requiring just one factor but A,B,C actually have 4 divisors rather than just one factor.

Limited Proof added 4/18/2015 at the same time Disproof #3 can be seen as a disproof by showing more than one common factor. A=2 x=3, B=2, y=3, C=2, z=4 can also be seen as a limited proof of a particular case as the equation has a case in which A,B, C all have the common prime factor of 2. Even one proof case could be seen as a proof meeting the Prize Requirements. There is one common prime factor of 2 here but also the 2 is not just one factor but rather 4 factors. A hypothesis as we see here can be both proven and disproven.

James T. STruck BA, BS, AA, MLIS

Disproofs added on 4/18/2015

1. A=1, B=1, C=1 x=3, y=3, z=3 then Beal's if statement equals 1+1=1 in which case the initial If statement is false raising questions about the conjecture or an if disproof. That is following the later conditions leads to a disproof of the initial If Condition. A,B,C in that case do not have a common prime factor also. An if statement can be disproven just as much as a then statement. Take the conditions, the If does not hold nor does then the statement as a type of disproof.

2. A=1, B=1, C=1, x=3, y=3, z=3 the Beal's statement equals 1+1=1 in which case the later condition is also disproved as 1 is not a prime number. Later conditions are met and even though the initial if condition is not met, the prime number requirement is also not met. To be a disproof, one can arguably disprove sections of an hypothesis or conjecture

3. Disproof A=0 x=4, B=0 y=5, C=0, z=6 then Ax + By = Cz,

but A,B, C at 0 do not have prime number factors as they are not prime numbers. 0 can arguably be seen as a positive integer as 0 is not negative

4. We can define A,B,C to be composite numbers that is numbers greater than 1 that are not prime. Take for example A=4, B=6, C=8. These are all composite numbers or numbers that have factor other than 1 and the number itself. If we then raise A,B,C to some theoretical x,yz, greater than 2 then we see again that A,B,C do not have a common prime factor as they were chosen to be numbers that are not prime numbers.

Proof Theory can be more complicated than it seems. We can take then then statement and show cases in which it does not work to disprove a conjecture. We can take later conditions to disprove the if Condition. We can take infinity as it includes the integers to disprove conjectures like here as it would include the positive integers but also numbers that are not prime. The whole proof system of we need if to be true can be reversed and we can take the then to disprove the If also. So take one common prime factor numbers like 2. 2 to the 3rd, 2 to the 4th does not equal 2 to the fifth. We have common prime factors of 2 middle positive integers and x,y,z greater than 2 conditions met, but we disprove the conjecture by showing that 24 does not equal 32. Similarly taking composite numbers or numbers that are not prime for A,B,C we have disproven the hypothesis using middle conditions of positive integers and also commonly the IF statement here. Disproofs do not have to follow the form "If cows were pink and green, then we would see them as pink and green so we have to now use pink and green cows." Instead we can show cases in which cows are not pink and green to disprove the if of a statement about a pink and green cow statement. Here again we do also show some limited proof of the conjecture by Beal as well.

James T. Struck shows proof and disproof can occur simultaneously and one can take then statements and use them to show If statements false or middle statements false serving as a type of disproof as well.

BEAL'S CONJECTURE: If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.

One Last Disproof. If Ax + By = Cz, where A, B, C,

BUT A,B,C are composite numbers or numbers that are not prime numbers or numbers that do not have one common prime factor, then the conjecture is also disproved as we set up conditions in which there would be no common prime factor!! Proof through choosing conditions in which the THEN statement is the disproved by requirement of the disproof or disproof by requirement!

We REQUIRE THAT A, B and C DO NOT have a common prime factor, disproving the conjecture by new conditions.

We can use disproofs or show disproofs by showing or setting up conditions in which the Then statement is required to be false to disprove math statements.

- Proof and Disproof at the Same Time-Beal Conjecture Case Study Proof 2015,
- Disproven 2015 and 2013 Beal Conjecture Disproven

*Travaux, Lithuanian, Illinois Journal of math, French Journal of Math, American Journal of Math*(2015)

Available at: http://works.bepress.com/james_struck/57/