Crux Mathematicorum with Mathematical Mayhem, is a problem solving journal published by the Canadian Mathematical Society. In the March 2010 issue(see reference[1]) ,the following problem was proposed:Determine all positive integers a,b, and c such that a^(b^c)=(a^b)^c; or equivalently, a^(b^c)=a^(b^c). A solution by this author was published in the December2010 issue of Crux(see reference[2]). Accordingly, all such positive integer triples are the following:The triples of the form (1,b,c); with b, c any positive integers; the triples (a,b,1); a, b positive integers, with a being at least 2; and the triples of the form (a,2,2); a being a positive integer not equal to 1.These are then the positive integer solutions to the 3-variable exponential diophantine equation, x^(y^z)=x^(yz) (1) Motivated by mayhem problem M429, in this work we investigate for more 3-variable exponential diophantine equations: x^(y^z)=x^(z^y) (2), x^(y^z)=y^(xz) (3) x^(yz)=y^(xz) (4), x^(y^z)=z^(xy) (5) We completely determine the positive integer solution sets of equations (2), (3), and (4). This is done in Theorems2,3, and4 respectively. We also find three different families of solutions to equation (5); listed in Theorem5.
In the March 2010 issue of the journal Crux Mathematicorum with Mathematical Mayhem, mayhem problem M429 was proposed (see reference [1]): Determine all positive integers a, b, c that satisfy, a (b c ) = (a b ) c ; or equivalently
A solution, by this author, was published in the December 2010 issue of Crux Mathematicorum with Mathematical Mayhem (see [2]). According to this solution, the following ordered triples of positive integers a, b, c are precisely those that satisfy the above exponentialequation:
The triples of the form (1, b, c), with b, c being any positive integers; the triples of the form (a, b, 1), with a, b positive integers and with a ≥ 2; and the triples of the form (a, 2, 2) with a ∈ Z + , and a ≥ 2.
In the language of diophantine equations, we are dealing with the threevariable diophantine equation
Accordingly, the above results can be expressed in Theorem 1 as follows.
Theorem 1. Consider the three-variable diophantine equation, x (y z ) = x yz , over the set of positive integers Z + . If S is the solution set of the above diophantine equation, then S = S 1 S 2 S 3 , where S 1 , S 2 , S 3 are the pairwise disjoint sets,
Motivated by mayhem problem M429, in this work we tackle another four exponential, three-variable diophantine equations. These are:
and
In Section 2, we state Theorems 2, 3, 4, and 5. In Theorems 2, 3 and 4, the solutions sets of the diophantine equations ( 2), (3), and (4) are stated.
These three solution sets are determined with the aid of the two-variable exponential diophantine equation found in Section 3, whose solution set is given in Result 2.
The proofs of Theorems 2,3, and 4, are given in Section 4. The proof of Theorem 5 is presented in Section 5. In Theorem 5, some solutions to equation (5) are given.
2 The four theorems Theorem 2. Consider the three-variable diophantine equation (over Z + ),
Let S be the solution set of this equation.
Then, S = S 1 S 2 S 3 S 4 S 5 , where
Consider the three-variable diophantine equation (over Z + ),
Let S be the solution set of this equation. Then, S = S 1 S 2 S 3 S 4 S 5 where
Let S be its solution set. Then,
where
Theorem 5. Consider the three-variable equation (over Z + )
x (y z ) = z xy (i) Let S be the set of those solutions, (x, y, z) such that at least one of x, y, or z is equal to 1. Then
The only solution (x, y, z) to the above equation, such that x ≥ 2, y ≥ 2, z ≥ 2, and with x = z, is the triple (2, 2, 2)
The diophantine equation, x y = y x , over the positive integers, is instrumental in determining the solution sets of the diophantine equations ( 2), (3), and (4).
The following, Result 1, can be found in W. Sierpinski’s book, “Elementary Theory of Numbers”, (see reference [3]). The proof is about half a page long.
Result 1. Consider the two-variable equation, x y = y x , over the set of positive rational numbers, Q + . Then all the solutions to this equation, with x and y being positive rationals, and with y > x, are given by
where n is a positive integer: n = 1, 2, 3, . . . .
A simple or cursory examination of the formulas in Result 1 easily leads to Result 2. Observe that these formulas can be written in the form,
For n = 1, we obtain the integer solution x = 2 and y = 4. However, for n ≥ 2, the number n + 1 n is a proper rational, i.e., a rational which is not an integer. This is clear since n and n + 1 are relatively prime, and n ≥ 2. Thus, since for n ≥ 2, n + 1 n is a proper rational, so must be any positive integer power of n + 1 n . This observation takes us immediately to Result 2 below.
Result 2. Consider the two-variable diophantine equation (over Z + )
Let S be its solution set. Then, S = S 1 S 2 S 3 . Where 4 Proofs of Theorems 2, 3, and 4
(1) Proof. Theorem 2 Suppose that (a, b, c) is a solution to equation (2). We have 3), then it must belong to one of the sets S 1 , S 2 , S 3 , S 4 or S 5 . Conversely, a routine calculation shows that any member of these five sets is a solution to (3).
(3) Proof. Theorem 4. Let (a, b, c) be a positive integer solution to equation (4) We have shown that if (a, b, c) is a positive integer solution of equation (4), it must belong to one of the sets S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , or S 7 . Conversely, a routine calculation establishes that any member of these seven sets is a solution to (4).
(5) Proof of Theorem 5 The following lemma can be easily proved by using mathematical induction. We omit the details. We will use the lemma in the proof of Theorem 5.
Lemma 1.
(ii)) 2 n-1 > n, for all positive integers n ≥ 3.
This content is AI-processed based on open access ArXiv data.