Is it a rational, irrational, even or odd

Question

If !!a^2+b^2+c^2=d!!, where !!a!! and !!b!! are consecutive positive integers and c=ab then !!sqrt(D)!! is?

  • Sometimes rational
  • Sometimes odd
  • Always odd

It is given !!a!!, !!b!! are consecutive numbers so without the loss generality we can assume !!b=a+1!!

So the given equation become !!D=a^2+(a+1)^2+a^2(1+a)^2!!

!!a^2(1+a)^2!! is always even. And exactly one of !!a^2!!, !!(1+a)^2!! is even. So !!D!! is odd. Which implies !!sqrtD!! will be an odd if it is an integer.

Now let us try to figure if it is perfect square.

!!D=a^2+(a+1)^2+a^2(1+a)^2!!

!!=2a(a+1)+1+a^2(1+a)^2!!

!!=(a(1+a)+1)^2!!

So D is a perfect square and an integer.

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