2005 iTest Problems/Problem 34

Problem

If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$ .

Solution 1

Let $a=x^2, b=(x+1)^2, c=(x+2)^2, d = y^2$ , where $x$ and $y$ are integers. Then we have the following result: $x^4+(x+1)^4+(x+2)^4=y^4$ . This equation has no solutions in the integers, so $\boxed{x=0}$ .

2005 iTest (Problems, Answer Key)
Preceded by: Problem 33	Followed by: Problem 35
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2005 iTest Problems/Problem 34

Problem

Solution 1

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