DIOPHANTINE EQUATIONS AND IDENTITIES

The general diophantine equations of the second and third degree are far from being totally solved The equations considered in this paper are i x2 my =+l ii) x3 +my3+m z3 3 mxyz i iii) Some fifth degree diophantine equations Infinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before It is known that the solutions of Pell’s equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell’s equations case. Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.


O. INTRODUCTION
In this paper we shall investigate Diophantine equations of the second and third degree of a special type.The general equations of the second and third degrees are far from being totally solved.It suffices to look up Mordell's book on Diophantine equations, to learn how little we actually know about the general second and third degree Diophantine equations, in spite of the many numerous results on this subject that have been gained by great mathematicians with no little effort.
The famous Thue theorem stating that the equation aoxn + alxn-ly + an_lxyn-1 n + anY c (a i,c rational integers, i 0,I,... ,n; n> 2) has only a finite number of (rational) solutions is an amazing dis- covery.It leaves open the question how to find these solutions and what is their exact number, and one would conjecture that it will remain open for (all) times to come.
The equations considered in this part of the paper are: i) The equation, known (wrongly) as Pell' s equation, namely 2 2 x -my :+i, ii) The equation x 3 +my3+m2z 3-3mxyz i, x Infinitely many solutions of each of these equations will be stated explicitly.Now, it is known that the solutions of Pell's equations is well exploited.
Still, we found it necessary to include it here because of the simple method we shall use in solving this equation here, which has such a wide range of application in various branches of exact sciences.Also, we will derive some new combinatorial identities.
The ACF [i] of a(O)=w+D is purely periodic with length of its primitive period $ i; hence we have from formula (0.5) n )n A(0n) w+D)A(on+l) (2.4) From (2.3) and (2.4), and using the expressions for X2n and Y2n from the previous paragraph, we obtain the combinatorial identities n-1 i=0 i=0 (2,5) Similar identities are obtainable from (w+D) 2n+l X2n+l + Y2n+l w.
As we shall soon see, there is a much simpler third degree Diophantine equation which can be regarded as, and indeed in a certain case represents, a generalization of Pell's equation to the third degree.

UNITS IN THE CUBIC FIELD
As we have seen in [i], the ACF of the vector a (0) E3, with D 2 w D e N, a (0) (w+2D w2+Dw+ ), is purely periodic with length of primitive period Z I. Hence, by theorem 2 in [2] and formula (0. I0) We shall find the field equation of the expressions (1.3) in Q(w).X3v + mYv3 + m z v 3mXvYvZv 1 Xv,Yv,Z v from (4.4), v 0,i, The Diophantine equation is indeed Pell's equation generalized to the third dimension.It is simpler compared with (3.6) and it has as solutions (4.4).We shall verify formula (4.5), first line for v 1,2.We have, from (4.4), z AO(5) Z.
We shall now extract a few interesting identities from Formula (4.3).

O<j<i<3n
For this purpose we have to (4.8) (4.8) is an appealing formula for the expression (w2+Dw+D2) 3n, though this expression could also be calculated by the multinomial theorem.
The coefficient of w in the expansion of (w2+Dw+D2) 3 equals as the reader can verify.
For n=O, x=l, y= z=u=v=O, the determinant in (6.5) becomes I but these elementary determinants can hardly serve as a verification for formula (6.5).For n 2 the test is also simple.
Expanding (8.2), with m=w5=D5+l, we obtain a5cI +malc2 + ma2c3 + ma3c4 + ma4c5 1 a4cI + a5c2+malc3+ma2c4+ma3c5 0 a3cI + a4c2+ a5c3+malc4+ma2c5 0 a2cI + a3c2+ a4c3+ a5c4+malc5 0 alcI + a2c2+ a3c3+ a4c4+ a5c5 O. (8. 2) The determinant of the system of linear equations (8.3)Now, the reader will verify that the field equation of e 5n a 5 + a4w + a3w2 + a2w3 + alw4 has exactly the free element =l, since e is a unit, as in case n 3. We thus obtain (w_D)5n Cl +c2w+c3w2 4 +c4w3+CsW Expanding (w-D) 5n we obtain the result.The rational part in the expansion of (8.For the right side we calculate Thus the determinant (8.8) becomes, with the values from (8.1), viz.On the combined subject of this paper about "Diophantine Equations, Units and Identities" there is not much literature, but I cannot finish without naming the literature in each of the three above mentioned sub- jects without indicating at the very end, some papers which have been most useful in my paper.
obtained the identity.

8 )
has been verified for D n 1.The entries in the right hand determinant become a challenge for n,D >1. 2n+2 equals, inter- changing columns with rows,