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The number of non-negative integers 'n' ...

The number of non-negative integers 'n' satisfying `n^2 = p+q` and `n^3 = p^2+ q^2` where p and q are integers
(A) 2
(B) 3
(C) 4
(D) infinite

Text Solution

Verified by Experts

Here, `n^2 = p+q`
`=>(n^2)^2 = (p+q)^2`
`=>n^4 = p^2+q^2+2pq`
`=>n^4 = n^3+2pq`...[As `n^3 = p^2+q^2`]
Now, arithmetic mean of two numbers is greater than that of geometric mean of those two numbers.
`:. (p^2+q^2)/2 ge sqrt(p^2q^2)`
`=> (p^2+q^2) ge 2pq`
`:. n^4 le n^3 + p^2+q^2`
...
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