Equal Sums of Like Powers

Integer Solutions of

• Chen Shuwen found the first solution of this type in 1997 .
• [ -372, 43, 371 ] = [ -405, 140, 307 ]
• Ajai Choudhry obtained a method of generating infinitely many integer solutions of this type in 1999 [61]. Numerical examples are:
• [ -300, 83, 211 ] = [ -124, -185, 303 ]
• This is also the smallest solution.
• [ -479, 23, 432 ] = [ -393, -127, 496 ]
• To the system ( the m=n case )
a₁^h + a₂^h + ... + a_n^h = b₁^h + b₂^h + ... + b_n^h      ( h = h₁ , h₂ , ... , h_n )
Integer solution had been obtained for 10 types of ( h = h₁ , h₂ , ... , h_n ) so far.
All these 10 types, except ( h = 1, 2, 6 ), can be express as ( h = 1, 2, ..., s, s+2, ..., s+2t )
• ( h = 1, 2, 4 )
• ( h = 1, 2, 6 )
• ( h = 1, 2, 3, 5 )
• ( h = 1, 2, 4, 6 )
• ( h = 1, 2, 3, 4, 6 )
• ( h = 1, 2, 3, 5, 7 )
• ( h = 1, 2, 4, 6, 8 )
• ( h = 1, 2, 3, 4, 5, 7 )
• ( h = 1, 2, 3, 4, 5, 6, 8 )
• ( h = 1, 2, 3, 4, 5, 6, 7, 9 )