# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k+ a4k = b1k + b2k + b3k + b4k
( k = 2, 3, 4 )
• H.Gupta had proved that the following system has no nontrivial solutions in positive integers. [8]
• a1k + a2k + ... + an-1k = b1k + b2k + ... + bn-1k      ( 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.
• T.N.Sinha conjectured that the system
• a1k + a2k + ... + ank = b1k + b2k + ... + bnk      ( 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.
• Here are some integer solutions of this type ( by Chen Shuwen)
• [ -26, 52, 93, 111 ] = [ 39, 58, 76, 117 ]
• [ -43, -3, 200, 215 ] = [ 32, 47, 185, 225 ]

Last revised March,31, 2001.