Equal Sums of Like Powers

Algebraic Identities

( The Generalization of Ramanujan's 6-10-8 Identity )

 


Identities for ( h = 1, 2, 4 )

Let   Fn = ( a + b + c )n + ( b + c + d )n -( c + d + a )n -( d + a + b )n +( a - d )n -( b - c )n

and ad = bc, then   

64 F6 F10 = 45 F82           ( Ramanujan )

25 F3 F7 = 21 F52            ( Hirschhorn )

5 F3 F8 = 8 F5 F6             ( Chamberland )

16 F6 F7 = 7 F3 F10           ( Chamberland )

F-2 F32 = -3 F-12 F6           ( Chamberland )

245 F3 F11 + 330 F72 = 539 F5 F9        ( Chamberland )

300 F6 F14 + 308 F102 = 525 F8 F12       ( Chamberland )

Let   Pn = (a1n + a2n + a3n - b1n - b2n - b3n ) / n

if  P1 = P2 = P4 = 0 , then   

4 P6 P10 = 3 P82   

P3 P7 = P52      

P3 P8 = 2 P5 P6

3 P6 P7 = P3 P10

P-2 P32 = -P-12 P6

P3 P11 + 2 P72 = 3 P5 P9

9 P6 P14 + 11 P102 = 18 P8 P12

Let   Pn = (a1n + a2n + a3n - b1n - b2n - b3n ) / n , and P0 = 2 ( a1 a2 a3 - b1 b2 b3 ) / (a1 a2 a3 + b1 b2 b3 )

if  P1 = P2 = P4 = 0 , then   

P5 / P3 = P7 / P5 = P8 / 2P6 = 2P10 / 3P8 = ( P7 / P3 )1/2 = ( P10 / 3P6 )1/2 = (a12 + a22 + a32 ) / 2

P8 / 2P5 = P3 / P0 = P6 / P3 = P10 / 3P7 = (a1 a2 a3 + b1 b2 b3 ) / 2

3P52 / P10 = P32 / P6 = -P-12 / P-2 = P0 = 2 * (a1 a2 a3 - b1 b2 b3 ) / (a1 a2 a3 + b1 b2 b3 )

and

P3 = a1 a2 a3 - b1 b2 b3

P5 = ( a1 a2 a3 - b1 b2 b3 ) (a12 + a22 + a32 ) / 2

P6 = ( a12 a22 a32 - b12 b22 b32 ) / 2

P7 = ( a1 a2 a3 - b1 b2 b3 ) (a12 + a22 + a32 )2 / 4

P8 = ( a12 a22 a32 - b12 b22 b32 ) (a12 + a22 + a32 ) / 2

P10 = 3( a12 a22 a32 - b12 b22 b32 ) (a12 + a22 + a32 )2 / 8

 


Identities for ( h = 1, 2, 4 ) , ( h = 0, 1, 3 ) , ( h = -1, 0, 2 ) , ( h = -2, -1, 1 )

Let   Pn = (a1n + a2n + a3n - b1n - b2n - b3n ) / n , and P0 = 2 ( a1 a2 a3 - b1 b2 b3 ) / (a1 a2 a3 + b1 b2 b3 )

 

 


Identities for ( h = 1, 3, 4 ) , ( h = 0, 2, 3 ) , ( h = -1, 1, 2 ) , ( h = -2, 0, 1 )

Let   Pn = (a1n + a2n + a3n - b1n - b2n - b3n ) / n , and P0 = 2 ( a1 a2 a3 - b1 b2 b3 ) / (a1 a2 a3 + b1 b2 b3 )

 

 


Identities for ( h = 2, 3, 4 ) , ( h = 1, 2, 6 )

Let   Pn = (a1n + a2n + a3n - b1n - b2n - b3n ) / n , and P0 = 2 ( a1 a2 a3 - b1 b2 b3 ) / (a1 a2 a3 + b1 b2 b3 )

 

 

 


Identities for ( h = 1, 2, 3, 5 ) , ( h = 0, 1, 2, 4 ) , ( h = -1, 0, 1, 3 ) , ( h = -2, -1, 0, 2 ) ,( h = -3, -2, -1, 1 )

Let   Pn = (a1n + a2n + a3n + a4n - b1n - b2n - b3n - b4n ) / n , and P0 = 2 ( a1 a2 a3 a4 - b1 b2 b3 b4 ) / (a1 a2 a3 a4 + b1 b2 b3 b4 )

 


Identities for ( h = 1, 2, 4, 6 ) , ( h = 0, 1, 3, 5 ) and ( h = -2, -1, 1, 2 )

Let   Pn = (a1n + a2n + a3n + a4n - b1n - b2n - b3n - b4n ) / n , and P0 = 2 ( a1 a2 a3 a4 - b1 b2 b3 b4 ) / (a1 a2 a3 a4 + b1 b2 b3 b4 )

 

 

P-3 P-5 / P-4 = P3 P5 / P4        ( Perfect !!! )

P6 = P52 / P4  + P32 / P0

P7 = P53 / P42 + 2 P3 P4 / P0

P8 = P54 / P43 + 2 P3 P5 / P0 + P42 / P0

 


Identities for ( k = 1, 3 ) , ( k = 1, 2, 4 ) , ( k = 1, 2, 3, 5 ) and ( k = 1, 2, 3, 4, 6 )

  if  R1 = R= R3 = R5 = 0 , then   R43 ( R4 R9 - 2 R6 R7 ) = ∏i5=1j5=1( ai - bj )

  if  R1 = R= R3 = R4 = R6 = 0 , then   R53 ( R5 R82 + 2 R5 R7 R9 - R11 R52 - R73 ) = ∏i6=1j6=1( ai - bj )

 


Identities for ( k = 1, 2 ) , ( k = 1, 3 ) , ( k = 1, 4 ) and ( k = 1, 5 )

   R1 S + R33 - 2R2 R3 R4 + R22 R5 = ∏i3=1j3=1( ai - bj )

    where S is a polynomial containing 9 items.


Identities for ( k = 1, 2 ) and ( k = 0, 1 ) , ( k = 1, 2, 3 ) and ( k = 0, 1, 2 ) , ( k = 1, 2, 3, 4 ) and ( k = 1, 2, 3, 4, 5 )

  if  R1 = R= R3 = R4 = 0 , then  

R5 R7 / R62 = ( m + 1 ) / 2 , where m = ( a12 + a22 + a32 + a42 + a52 ) / ( a1 + a2 + a3 + a4 + a5 )2

  if  R1 = R= R3 = R4 = R5 = 0 , then   

R6 R8 / R72 = ( m + 1 ) / 2 , where m = ( a12 + a22 + a32 + a42 + a52 + a62 ) / ( a1 + a2 + a3 + a4 + a5 + a6 )2

 


Reference for this page


Last revised Dec,7, 2017.
Copyright 1997-2017, Chen Shuwen