SELF

54

S.B. Karavashkin and O.N. Karavashkina

If   a  is the divisor of B ² , the condition (8) changes so:

With it we suppose that a  is the square of some integer number. Given (11), the expression (7) will take the following shape:

In (13), in order to keep A  integer, if we take k  even, we have to take a  also even, and vice versa.

At

(7) has the following solutions for the involved positive unknowns:

that basically differ from the solutions (10).

In this small investigation we can see, how the solution for a simple case x = y = z = 2  branches, though the general approach remains and can be extended for more complex versions of the equation (1). We can easily prove that, representing C  in the form (5), we will always come to the branched solutions also for more general condition

Actually, substituting (5) into

we come to the equation of the type

(where  C_n^j   is the binomial factor), whence it follows that  a  has to be the divisor of B ⁿ . With it, there appears more general branching of the solution, which is determined by the condition

where  a₁   is the integer number. Dependently on p , there will exist a spectrum of different solutions determined by the solution of the power-type equation

At inequal values x, y, z  the above technique in general case already doesn't work, and we have to approach otherwise, dependently on the relation between  A, B, C . The equation (1) determines that this relation exists, and at least one of unknowns can be found through two others. Just this has required to introduce the additional conditions when we presented the technique for equal powers. The same conditions are required also in general case, as without them we cannot select the range of numbers in which the solution is located.

Specifically, if A ^x   is the integer divisor of B ^y , we can represent B ^y as

With it (1) will take the following form:

It follows from (22) that (21) gives rise to the condition

It means, any pairs of numbers B₁  and C₁ satisfying (24) lead to the solution (1) under condition (21). As we can see, such approach to the solution basically differs from the approach when the powers were equal. True, we can show that under definite conditions the expression (24) is the generalisation of solution (15). Actually, if in (15) we choose A  and C  so that we could represent them in the form

Subtracting in (26) the second equality from that first, we yield

At k₁ = k₂ = 1  it reduces to (24), but in comparison with (24) there arises also a difference connected with the fact that a in general case is not equal to 1. Proceeding from this, we can record the general condition on the basis of (24) and (27) as

This condition is important when seeking the general solution, because it contains only two unknowns of three (A, B, C), while the third unknown is substituted for the given parameters a , k₁  and k₂ . Furthermore, in particular case (27) we can arbitrarily give A₁, C₁, p, q and on the basis of relationship (26) to determine  a   and k . In fact we so will determine B . The obtained solution will satisfy the equation

If with it we choose k  so that to satisfy the condition

we will yield more general expression

limited by the conditions (30).

The presented analysis shows that there exists no general technique to solve the equation (1). Dependently on relations between the involved unknowns, the approaches to the solution will change and evidence the multiple branching which is typical for the number theory. Just this we showed in our analysis.