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Triples of Positive Integers (RMO 2015 Problem 3)

Problem:  Show that there are infinitely many triples $$(x,y,z)$$ of positive integers, such that $$x^3+y^4=z^{31}$$.

Discussion

Suppose we have found one such triplet (x, y, z). Then $$x^3 + y^4 = z^{31}$$. Multiply $$a^{372}$$ to both sides where a is an arbitrary integer.

Clearly we have $$a^{372}x^3 + a^{372}y^4 = a^{372}z^{31}$$

$$\Rightarrow (a^{124}x)^3 + (a^{93}y)^4 = (a^{12}z)^{31}$$

Hence if (x, y, z) is a triple then $$(a^{124}x, a^{93}y, a^{12}z )$$ is another such triple. Since a can be any arbitrary integer, hence we have found infinitely many such triplets provided we have found at least one

To find one such triple, we use the following intuition: set x, y, z as some powers of 2 such that $$x^3 = y^4 = 2^{r}$$. Then r must be of the form 12k. Finally, their sum must be $$x^3 + y^4 = 2^{r} + 2^{r} = 2^{r+1}$$. This r+1 must be divisible by 31.

Let $$r = 12s$$ and $$r+1 = 31m$$ we get $$12s +1 = 31m$$. Since 12 and 31 are coprime there is integer solution to this linear diophantine equation (by Bezoat’s theorem). We can solve this linear diophantine equation by euclidean algorithm.

$$31 = 12 \times 2 + 7$$
$$12 = 7\times 1 + 5$$
$$7 = 5\times 1 + 2$$
$$5 = 2 \times 2 + 1$$
$$\Rightarrow 1 = 5 – 2 \times 2 = 5 – 2 \times (7 – 5 \times 1)$$
$$\Rightarrow 1 = 3 \times 5 – 2 \times 7 = 3 \times (12 – 7 \times 1) – 2 \times 7$$
$$\Rightarrow 1 = 3 \times 12 – 5 \times 7 = 3 \times 12 – 5 \times (31 – 12 \times 2)$$
$$\Rightarrow 1 = 13 \times 12 – 5 \times 31$$
$$\Rightarrow 1 = 13 \times 12 – 5 \times 31 + 12 \times 31 – 12 \times 31$$
$$\Rightarrow 1 = (13 -31)\times 12 +(12- 5 )\times 31$$
$$\Rightarrow 1 = -18 \times 12 + 7\times 31$$
$$\Rightarrow 1 + 18 \times 12 = 7\times 31$$
Hence we use this to form an equation:
$$2^{18 \times 12} + 2^{18 \times 12} = 2^{216} + 2^{216} = 2^{217}=2^{7\times 31}$$
$$(2^{72})^3 + (2^{54})^4 = (2^7)^{31}$$

Hence we have found one such triple : $$(2^{72}, 2^{54} ,2^7 )$$ (from which we have shown earlier that infinitely more can be generated)

Chatuspathi:

• What is this topic: Number Theory
• What are some of the associated concept: Linear Diophantine Equation, Bezoat’s Theorem, Euclidean Algorithm, Sum of powers of two
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, Cheenta Math Olympiad Program, discuss these topics in the ‘Number Theory’ module.
• Book Suggestions: Elementary Number Theory by David Burton
December 7, 2015

7 comments

1. I found, in some question papers, it is mentioned only ‘integers’ not ‘positive integers’. I know, that it should be ‘positive integers’, otherwise it will become a trivial problem by taking x=0.

• That is right 🙂

2. But in original question (x,y,z) were said to be only integers; not necessarily positive.

• Hmm… I think the intension was positive integer solution.. Otherwise the problem becomes trivial..

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