RMO 2015

Number Theory Level 3

How many triplets of integers ( x , y , z ) (x,y,z) exist such that they satisfy the equation below?

x 3 + y 4 = z 31 \large{x^3+y^4=z^{31}}

Can't be determined 0 Infinite 358 38

Divyansh Chaturvedi by Divyansh Chaturvedi , 21, India

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1 solution

Let us take a special case in which x is 0

So we have

y⁴ = z³¹

Since both are integers z is of the form k⁴ (in simplest form) Therefore y will be equal to k³¹ Hence we get the triplet (0,k³¹,k⁴)

Now observe that there are infinite triplets as for every different internal value is k we get a unique triplet..... Hence the ans is infinite.. Since this is only the case with x=0 Other cases which were left will only add up to the list......

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You could have also shown that x = − m 4 , y = m 3 x=-m^4, y=m^3 for some integer m m .

Department 8 - 5 years, 5 months ago

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But isn't that the same thing ?? I put x=0 U putting z=0 But yeah its also a way to do .....

Divyansh Chaturvedi - 5 years, 5 months ago

RMO problem I see ;)

Rwit Panda - 5 years, 5 months ago

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