`Equal Sums of Like Powers`

Non-negative Integer Solutions of

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

( k = 2, 3, 4 )

H.Gupta had proved that the following system has no nontrivial solutions in positive integers. [8]

a₁^k + a₂^k + ... + a_n_-1^k = b₁^k + b₂^k + ... + b_n_-1^k ( k = 2, 3, ..., n )

Non-negative integer solutions of this type were first obtained by Chen Shuwen in 1995.

[ 975, 224368, 300495, 366448 ] = [ 37648, 202575, 337168, 344655 ]
[ 7001616, 10868299, 31439172, 34940503 ] = [ 7527024, 10393591, 31599228, 34831147 ]
[ 2756106, 17971525, 31568076, 35616295 ] = [ 3727405, 17323956, 32539375, 34968726 ]
[ 33801840, 3033353281, 4414180500, 5723026141 ] = [ 1004104381, 2384931600, 5074604460, 5384483041 ]

Chen found the above results with the help of a 386SX/33 PC.

a₁^k + a₂^k + ... + a_n^k = b₁^k + b₂^k + ... + b_n^k ( k = 1, 2, ..., j-1 , j+1, ..., n )

has nontrivial solution in positive integer for all n. However he could only prove the cases n<=3. [6] [32]

Now with the following results obtained by Chen, we can prove that T.N.Sinha's conjecture is true for n<=4.

[ 2, 7, 11, 15 ] = [ 3, 5, 13, 14 ] ( k = 1, 2, 4 )
[ 3, 140, 149, 252 ] = [ 50, 54, 201, 239 ] ( k = 1, 3, 4 )
[ 975, 224368, 300495, 366448 ] = [ 37648, 202575, 337168, 344655 ] ( k = 2, 3, 4 )