# Equal Sums of Like Powers

## Non-negative Integer Solutions of

a1k + a2k + a3k+ a4k = b1k + b2k + b3k + b4k
( k = 1, 3, 7 )
• Ajai Choudhry obtained solutions of this system in early 1999 [60] . Here is one of the numerical solution:
• Two methods are sucessfully used almost at the same time in Sep1999. One is programmed by Jean-Charles Meyrignac, search by Nuutti Kuosa, and noticed by Chen Shuwen. Another is by Chen Shuwen. Both obtained the following smallest result of this type.
• [ 184, 443, 556, 698 ] = [ 230, 353, 625, 673 ]
• Chen Shuwen wrote a program which can search all the solutions of ( k = 1, 3, 7 ) in a certain range and discovered the above solution in 10 Sep1999 after a Pentium 100M running 68 hours. Then, Chen Shuwen found that Nuutti Kuosa had obtained the following result in 8 Sep1999 by using a program that Jean-Charles Meyrignac wrote. Jean-Charles Meyrignac's program can search all the solutions of ( 7, 4, 4 ) in a certain range.
• Comparison of the two methods for ( k = 1, 3, 7 )
•  Jean-Charles Meyrignac's Methods Chen Shuwen's Methods Searched by Nuutti Kuosa, noticed by Chen Shuwen Searched by Chen Shuwen Found solution in 10 Sep1999 Found solution in 10 Sep1999 Can search all solutions of (7,4,4), include ( k=1,3,7 ) Only can search all solutions of ( k=1,3,7 ) CPU: PIII-450MHz, RAM: 768 MB CPU: PI-100MHz, RAM: 24MB Program language: ASM Program language: VB 5.0 Computing time: 3 days? ( Range: 1 to 698) Computing time: 68hours ( Range: 1 to 698) Computing speed: 1162sec ( Range: 698 to 698) Computing speed: 1696sec ( Range: 698 to 698)

Last revised March,31, 2001.