Beal Conjecture Original Proved. Abstract

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Adele Nash
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1 Copyright by A. A. Frempong Beal Conjecture Original Proved "% of the people think; 10% of the people think that they think; and the other 8%would rather die than think."----thomas Edison "The simplest solution is usually the best solution"---albert Einstein Abstract Using a direct construction approach, the author proved the original Beal conjecture that if Ax + By = C, where ABCxy,,,,, are positive integers and xy,, > 2, then A, Band C have a common prime factor. The proof would be complete after showing that if A and B have a common prime factor, and C can be produced from Ax + By. In the proof, one begins with Ax + By and change this sum to the single power, C as was done in the preliminaries. It was determined that if Ax + By = C, then The proof is very simple, and occupies a single page. 1

2 Preliminaries Ax + By = C A = Dr, B = Es, and C = Ft ( Dr) x + ( Es) y =. Note that r, sand t are prime numbers Case 1: Let r, s and t be prime factors of A, Band C.respectively, where D, Eand F are If D= 1, E = 1, F = 1 Then, rx + sy = t Also A= r, B= s, and C = t If A, Band Chave a common prime factor, then it is necessary that Aand B have a common prime factor. Example 1: = 2 = ( 1 + ( 1 = ( 1 One will show that another name for is 2. We will write the sum on the left-hand side as a single power. If the sum has a common prime factor, 2, then 2 has the common prime factor, 2. Step 1: We will work on the two terms on the left, and change their sum to the term on the right. The two terms 2 and 2 have the common prime factor 2.. Now, if by operating on 2 and 2 together, if we obtain 2, then surely 2 has a common prime factor as the sum of 2 and 2 since 2 was obtained from the sum 2 and 2,, Factor out the greatest common factor = 2( 1+ 1) = 2( = 2 It is interesting how the "(1+1)" provided the much needed 2. Step 2: Since it has been shown above that = 2 ( 1 + 1) = 2 ( = 2 The 2 must has a common factor as 2 and 2, from which it was obtained.. From above, the common prime factor is 2, and 2

3 Case 2: Let r, s and t be prime factors of A, Band C.respectively, where D, Eand F are If D= 1, E = 1, F 1 Then, rx + sy = Also A= r, B= s, and C = Ft Example = 98 = ( 1 7) 6 + ( 1 7) 7 = ( 1 7) Step 1 :We will work on the two terms on the left, and change their sum to the term on the right. Inspection shows that the two terms 76 and 77 have the common prime factor 7.. Now, if by operating on the sum , we obtain 98,then we can conclude that all the three terms 76, 77 and 98 have a common prime factor, since 98 was obtained from the sum Factor out the greatest common factor = 7 6 ( 1+ 7) = 76() 8 = 76( = ( 7 ( = ( 72 = ( 9 = ( 98) = 98 It is interesting how the "(1+ 7 )" provided the much needed 2. Step 2: It has been shown that = 76( 1 + 7) = 76( 8) = 76( = ( 7 ( = ( 72 = ( 9 = ( 98), Since 98 was obtained from the sum , which has a common prime factor. 7, 98 has the same common prime factor, 7, Therefore A, Band C have a common factor.

4 Case : Let r, s and t be prime factors of A, Band C.respectively, where D, Eand F are If D= 1, E 1, F = 1 Then, rx + ( Es) y = t Also A= r, B= Es, and C = t Example : + 6 = = ( 1 ) + ( 2 ) = ( 1 ) Step 1: We will work on the two terms on the left, and change their sum to the term on the right. Inspection shows that the two terms and 6 have the common prime factor.. Now, if by operating on the sum + 6 together, we obtain,we can conclude that all the three terms, 6 and have the common prime factor, since the term on the right was produced from the two terms which have the common factor, Write the sum on the left-hand side as a single power Step 1: Factor out the greatest common factor. + 6 = + ( = + 2 = ( 1+ = ( 1+ 8) = ( 9) = 2 = It is interesting how the "(1+ 8 )" provided the much needed 2. Step 2: It has been shown that + 6 = + ( = + 2 = ( 1 + = ( 1 + 8) = ( 9) = 2 =, + 6 = From above, the common factor is, and A, Band C have a common factor.

5 Case : Let r, s and t be prime factors of A, Band C.respectively, where D, Eand F are If D= 1, E 1, F 1 Then, rx + ( Es) y = Also A= r, B= Es, and C = Ft Example = = ( ( = ( 2 Show that equals.write the sum on the left-hand side as a single power Step 1: We will work on the two terms on the left, and change their sum to the term on the right. Inspection shows that the two terms 29 and 8 have the common prime factor 2. Now, if by operating on the sum together, we obtain, we can conclude that all the three terms 29, 8 and have a common prime factor, since the term on the right was produced from the two terms on the left; and the two terms have a common prime factor. Factor out the greatest common factor = 29 + ( = = 29( 1+ 1) 29 It is interesting how the "(1+ 1)" provided = 2 the much needed 2. = 210 = ( 2 = ( ) = Step 2: It has been shown that = 29 + ( = = 29( 1+ 1) = 29 2 = 210 = ( 2 = ( ) =, = From above, the common factor is 2, and A, Band C have a common factor.

6 Case : Let r, s and t be prime factors of A, Band C.respectively, where D, Eand F are If D 1, E 1, F 1 Then, ( Dr) x + ( Es) y = Also A = Dr, B = Es, and C = Ft Example + 1 = 8= ( 2 17) + ( 17) = ( 17) Write the sum + 1 as a single power. Step 1: We will work on the two terms on the left, and change their sum to the term on the right. Inspection shows that the two terms and 1 have the common prime factor, 17.. Now, if by operating on and 1 together, we obtain 8, we can conclude that all the three terms, 1 and 8 have a common prime factor, since the term on the right was produced from the two terms on the left with the common factor, 17. Write the sum + 1 as a single power ( 17 + ( 17 ) ( ) 17( ) 17( ) 17 It is interesting how the = ( ) = ( provided the much needed = 17( magic ) = ( 17 ) 62 =. = 8 Therefore, + 1 = 8 Step 2: Since it has been shown that + 1 = ( 17 + ( 17 ), = 17( ) = 17( ) = 17( 6 = 17( ) = ( 17 ) = = 8, 8 was obtained from and 1 which have the common prime factor, 17, A, Band C have a common factor. 6

7 Example 6: Given 9 + = 11 Write the sum 9 + as a single power. Step 1: We will operate on the two terms on the left, and change their sum to the term on the right. Inspection shows that the two terms 9 and have the common prime factor.. Now, if by operating on 9 and together, we obtain 11,we can conclude that all the three terms 9, and 11 have a common prime factor, since the term on the right was produced from the two terms on the left. Write the sum 9 + as a single power 9 + = 9 + ( 9 6) = 9 + ( = 9 + ( = = 9( 1+ = 9( 1+ 8) = 9( 9) = 9 2 = 11 It is interesting how the provided the much needed 9.. Step 2: Since it has been shown that 9 + = 9 + ( = = 9( 1+ = 9( 1+ 8) = 9( 9) = 9 2 = 11, 9 + = 11, and 11 was obtained from 9 and which have the common prime factor,,. From above, the common factor is, and A, Band C have a common factor. 7

8 Example 7: + 66 = 6 Write + 66 as the single power of. We will work on the two terms on the left, and change their sum to a single power Inspection shows that the two terms and 66 have the common prime factor. Now, if by operating on and 66 together, we obtain 6 we can conclude that all the three terms, 66 and 6 have a common prime factor, since the term on the right was produced from the two terms on the left Step 1: Factor the sum on the left-hand side + 66 = ( 11 ) + ( 11 2 ) = = 11 ( 1 + = ( 11 ) ( 1 + = ( ) = 6 It is interesting how the provided the much needed Step 2: It has been shown that + 66 = + ( = + 2 = ( 1 + = ( ) = 6, + 66 = 6 From above, there are two common prime factors, and 11. and therefore, A, Band C have a common prime factor. 8

9 General Proof Given: Ax + By = C, ABCxy,,,,, are positive integers and xy,, > 2. Required: To prove that Plan: A necessary condition for A, Band C to have a common prime factor is that Aand B must have a common prime factor. The proof would be complete after showing that If A and B have a common prime factor, and C has been produced from Ax + By. Proof: Let r be a common prime factor of A and B. Then A= Dr, and B= Er., where D and E are positive integers. Also let t be a prime factor of C. Then C = Ft, where F is a positive integer. On begins with ( Dr) x + ( Er) y and change this sum to the single power, C = as was done in the preliminaries. ( Dr) x + ( Er) y = ( Dr ( Es) y ) x[ 1 + ( Dr) x ] (Factoring out the ( Dr) x ) = ( Dr ) [ ( Ft ) ( Es) y x + ( Dr) x ] (( Ft ) = 1, applying the substitution axiom + = ( ) [ ( Ft ) ( Dr ) x ( Ft ) Dr ( Es ) y x ] Dr) x (Adding the terms within the brackets) ( Dr) x+ ( Es) y = (canceling out the ( Dr) x ) [ ( Dr) x+ ( Es) y ] = (Factoring out ) = ( Dr) + ( Er) = ( Fr) = x y Since C = was obtained from Ax = ( Dr) x and By = ( Er) y which have the common prime factor r, C also has the common prime factor, r. and one can write ( Dr) x + ( Er) y = ( Fr), where t = r. Therefore, PS Other proofs of Beal Conjecture by the author are at vixra: ; vixra: , vixra: Adonten 9

Beal Conjecture Original Proved. Abstract

Copyright by A. A. Frempong Beal Conjecture Original Proved "% of the people think; 10% of the people think that they think; and the other 8%would rather die than think."----thomas Edison "The simplest