## Integer Solutions
of

*a*_{1}^{h}
+ a_{2}^{h} + a_{3}^{h}
+ a_{4}^{h}+ a_{5}^{h}
= b_{1}^{h} + b_{2}^{h}
+ b_{3}^{h} + b_{4}^{h}
+ b_{5}^{h}
( h = 1, 2, 4, 6,
8 )
- 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 a
^{2}+12b^{2}=r^{2},
12a^{2}+b^{2}=t^{2} .

- 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.)

*Last revised March,31, 2001.*

Copyright 1997-2001, Chen Shuwen