User talk:Advocateankursaini

CONJECTURE If A^x + B^y = C^z, 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.

SOLUTION

The above conjecture is a form of constructive algebraic equation which has infinitely many solutions for eg. •	The case x = y = z is Fermat’s last theorem, proven to have no solutions by Andrew Wiles in 1994.[6] •	The case gcd(x,y,z) > 2 is implied by Fermat's Last Theorem. •	The case x = 2, y = 4, and z = 4 was proven to have no solutions by Pierre de Fermat in the 1600s. (See one proof here.) •	The case (x, y, z) = (2, 3, 7) and all its permutations were proven to have only four solutions, none of them involving an even power greater than 2, by Bjorn Edward F. Schaefer, and Michael Stoll in 2005. [7] •	The case (x, y, z) = (2, 3, 8) and all its permutations are known to have only three solutions, none of them involving an even power greater than 2.[1] •	The case (x, y, z) = (2, 3, 9) and all its permutations are known to have only two solutions, neither of them involving an even power greater than 2.[1][8] •	The case x = 2, y = 3, and z = 10 was proved by David Brown in 2009.[9] •	The case x = 2, y = 4, and z > 3 was proved by Michael Bennet, Jordan Ellenberg, and Nathan Ng in 2009. [10] •	The case (x, y, z) = (n, n, 2) is known to have no solutions for n equal to 6, 9, or any prime number greater than 3.[1] •	The case x = y = n and z = 3 has been proven for n = 4 and n equal to any prime ≥ 3.[1] •	The case x = y = 3 has been proven for z = 4, z = 5 and 17 ≤ z ≤ 10000.[1] •	The case x = z = 4 has been proven for all y.[1] •	The case A = 1 is implied by Catalan’s Conjecture, proven in 2002 by Preda Mihăilescu. •	It is known that for every specific choice of exponents (x,y,z) there are at most finitely many solutions.[11] This is implied by Falting’s theorem. •	The abc conjecture, if true, implies that there are at most finitely many counterexamples to Beal's conjecture. •	Peter Norvig, Director of Research at Google, reported having conducted a series of numerical searches for solutions to the Beal conjecture. Among his results, he excluded all possible solutions having each of x, y, z ≤ 7 and each of A, B, C ≤ 250,000, as well as possible solutions having each of x, y, z ≤ 100 and each of A, B, C ≤ 10,000. (Source:WIKIPEDIA.ORG) http://en.wikipedia.org/wiki/Beal's_conjecture

To prove the conjecture we need to precisely maintain the meta- language of the meta-theory which remains connected to the conjecture.

To prove this conjecture we need to put a function in the system to generate an operation which in the end when compared with the principle equation, fulfills each aspect of the principal equation ( BEAL CONJECTURE ).

EVALUATION OF A FUNCTION

In construction of a function, one thing must be remembered that simple rules of mathematics need to be followed i.e. addition, subtraction, multiplication and division cautiously, Neverethess with the end result of the operation the following six 6 conditions need to be fulfilled to at least stand for a PROOF- Elements (A,B,C,x,y and z>+1) generated in the operation must be positive integers. #1 Elements (x,y,z) must be equal to or more than 3. #2 Elements (A,B,C) must have a common prime factor. #3 x+y ≠ z.                                                                                                         #4 and A+B ≠C                                                                                                   #5 A≠B≠C≠x≠y≠z. #6

Let k be the principal equation(A SET) i.e. A^x + B^y= C^z.

And operation to the f(k) is 2^13.

ASSERTION

This question has possible result when,
 * The assertion is based on simple addition and multiplication rule.
 * Even number here means positive even numbers excluding 0.

The thirteenth power of first even number is equal to twice the multiple of first and second even number with 4 as the exponents to both numbers respectively.

This means,''' (α)^13= 2[(α)^4*(β)^4]''' Here, α=2, first even number. And,   β=4, second even number. If we elaborate this theorem, Here, 2[{(2)^4}*{(4)^4}]=2^13,(1)

Now we can also write 2 as (1+1) Replacing the value of 2 in equation number (1). Thus, [1+1] [{(2)^4}*{(4)^4}]=2^13 ,[ {(2)^4}*{(4)^4}+{(2)^4}*{(4)^4} ]=2^13

Now we will take 4^4 as a common, 4^4 [2^4+2^4]=2^13--(2)

We know that 2^4=4^2=16, thus we will now replace 2^4 to 4^2 in equation nos. (2).

4^4 [ 4^2+4^2 ] = 2^13 =>[ 4^4*4^2]+[4^4*4^2]= 2^13 =>[(4^2)^2*4^2]+[4^6}= 2^13 =>[(16)^2*16]+4^6= 2^13


 * 16^3+4^6=2^13-(3)

Now equation nos 3 is a result of the operation Thus ,f(k) is f(A,B,C,x,y,z). Comparing equation nos 3 and the principal equation for which f(A,B,C,x,y,z) is a function. A^x + B^y = C^z and 16^3+4^6=2^13. Here,f(A,B,C,x,y,z)=f(k)=(16,4,2,3,6,13) Now putting Elements(A,B,C,x,y,z) i.e. (16,4,2,3,6,13) from the above operation to be put in A SET i.e A^x + B^y = C^z -	16^3+4^6=2^13 -	4096+4096=8192 -	8192=8192 Both the sides of the equation are equal thus this shows PRIMA FACIE the operation is conditionally correct unless other conditions from(#1---#6) are fulfilled.

Now if we see (#1--#6)- 	#1- (A,B,C,x,y,z)( 16,4,2,3,6,13)>1 	#2-(x,y,z)(3,6,13)>=3. 	#3-(A,B,C)must have a common prime factor here (16,4,2)has a common prime factor 2. 	#4-(x+y≠z) here 3+6≠13. 	#5-(A+B≠C) here (16+4≠2). 	#6-( A≠B≠C≠x≠y≠z) here(16≠4≠2≠3≠6≠13). Since points from #1--#6 are all proved once we put a function(A,B,C,x,y,z) to f(16,4,2,3,6,13) Thus the set of numbers (A,B,C,x,y,z) generated from the operation (16,4,2,3,6,13) qualifies for a potential proof of the conjecture.

''' FORMAL PROOF

''' Formal proofs are sequences of sentences. For a sentence to qualify as part of a proof, it has to be the product of applying an inference rule on previous ABC..xyz’s in the proof sequence. The last ABC..xyz’s in the sequence is recognized as a theorem. The point of view that generating formal proofs is all there is to mathematics is often called Formalism. Any language that one uses to talk about a formal system is called a meta-language. The meta-language may be nothing more than ordinary natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question. Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all ABC..xyz’s for which there is a proof.

Hence, to prove the conjecture with the above operation the elements generated out of the operation are different from the previous assumptions. Moreover it satisfies all the requirements from 1 to 6. Therefore this is the unique combination of (A,B,C,x,y,z) as ( 16,4,2,3,6,13) all positive integers that proves this conjecture is true in its very nature if we put an operation to 2^13, which once compared to principal equation satisfies all six condition. Thus for a proof, above operation is perfect.