`Equal Sums of Like Powers`

Non-negative Integer Solutions of

a₁ a₂ a₃ a₄ = b₁ b₂ b₃ b₄

a₁^k + a₂^k + a₃^k + a₄^k = b₁^k + b₂^k + b₃^k + b₄^k

( k = 0, 2, 3 )

By searching for the positive integer solutions of c₁^k + c₂^k+ c₃^k = d₁^k + d₂^k + d₃^k ( k = 2, 3 ), Chen Shuwen obtained the first solution for ( k = 0, 2, 3 ) in 13 Oct 2022.

[ 855, 4338, 10406, 16335 ] = [ 1215, 2838, 11495, 15906 ]

Among 2525 solutions of ( k = 2, 3 ) in the search range of {c_1,c_2,c_3,d_1,d_2,d₃}<=2473, Chen Shuwen noticed that [ 285, 1045, 1446 ] = [ 405, 946, 1485 ] ( k = 2, 3 ) also satisfies 285/1045=405/1485, then got the first solution for ( k = 0, 2, 3 ).

No solution was found in the range of {a_1,a_2,a_3,a_4,b_1,b_2,b₃,b₄}<=2015, by Chen Shuwen's computer search in 2022.

Non-negative Integer Solutions of

a₁ a₂ a₃ a₄ = b₁ b₂ b₃ b₄

a₁^k + a₂^k + a₃^k + a₄^k = b₁^k + b₂^k + b₃^k + b₄^k

( k = -3, -2, 0 )

Based on the above solution of ( k = 0, 2, 3 ) and Theorem 9, Chen Shuwen obtained the first solution of ( k = -3, -2, 0 ) in 13 Oct 2022.

[ 3544146, 5563485, 13345695, 67711842 ] = [ 3639735, 5036418, 20399445, 47649074 ]

Last revised October 13, 2022.
Copyright 1997-2022, Chen Shuwen