9nice one
platinum5 are u another b555 ???( or multiple identity of some other user)
edit ur statement abv
uve made a typo in powers in RHS
Prove that, for any integers a, b, c, there exists a positive integer n such
that \sqrt{n^{3}+an^{2}+bn+c} is not an integer.
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3 Answers
1let us assume thatn^{3}+an^{2}+bn+c is a sqaure for all n>0.
for sufficiently large `n` ,
\left(n^{\frac{3}{2}}+\frac{1}{2}an^{\frac{1}{2}}-1 \right)^{2}<(^{3}n+an^{2}+bn+c)<(n^{\frac{3}{2}}+\frac{1}{2}an^{\frac{1}{2}+1})^{2}
so if `n` is a lrge even perfect square, we have n^{3}+an^{2}+bn+c=\left(n^{\frac{3}{2}}+\frac{1}{2}an^{\frac{1}{2}} \right)^{2}
but RHS is not a perfect square for `n ` which is an even non square
91edit : in that step with 2 simultaneous inequalities, in the extreme right , the term is \left(n^{3}+\frac{1}{2}an^{\frac{1}{2}}+1\right)^{2}
