Equal Sums of Like Powers

Integer Solutions of

• It is clear that integer solutions of ( h = 1, 2, 4, 6, 8 ) can be translated into non-negative integer solutions of ( k = 2, 4, 6, 8 ) and ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 ) by using Theorem 4 .
• A.Letac obtained an ingenious method of this type, and gave the following solution in 1940's. [5] [25]
• [ -12, -20231, 11881, 20885, 23738 ] = [ -20449, 436, 11857, 20667, 23750 ]
• G.Palama obtained the second solution of this type in 1950 by using Letac's method. [33]
• [ 308520455907, -87647378809, 527907819623, -243086774390, 441746154196 ] = [ 432967471212, -338027122801, 529393533005, 133225698289, 189880696822 ]
• In 1966, T.N.Sinha translated Letac's method into the following form: [6] [34]
[ a-r, a+r, 4a, 3b-t, 3b+t ] = [ b-t, b+t, 4b, 3a-r, 3a+r ]      ( h = 1, 2, 4, 6, 8 )
here a²+12b²=r², 12a²+b²=t² .
• T.N.Sinha also proved that there exist an infinite number of distinct solutions by Letac's method.
• By using elliptic curve thoery, in 1990, C.J.Smyth proved that Latac's method produces infinitely many genuinely different solution[22]. He also gave an example as same as that by G.Palama.
• A note on this type:
• C.J.Smyth did not known the results of G.Palama[33] and T.N.Sinha[34] until his famous paper [22] published. (Personal communication between C.J.Smyth and Chen Shuwen in Jun.1995 and Dec.1995.)
• T.N.Sinha did not seen the paper of G.Palama[33] and C.J.Smyth[22] until 1996.(Personal communication between T.N.Sinha and Chen Shuwen in Aug.1995 and Sept.1996.)