kaliprasad said:Find n such that both $n+3$ and $n^2+3$ are perfect cubes
[sp]So if $n+3$ and $n^2+3$ are cubes then so is $(n+3)(n^2+3) = n^3 + 3n^2 + 3n = (n+1)^3 - 1$. But two consecutive numbers cannot both be cubes unless they are $0$ and $1$. Since $n^2 + 3 \geqslant 3$, the pair $0$ and $1$ cannot arise in this way. Therefore the question is asking for something impossible (which I think is a bit sneaky! (Tongueout) ).[/sp]kaliprasad said:hint
product of 2 cubes is a cube
Opalg said:[sp]So if $n+3$ and $n^2+3$ are cubes then so is $(n+3)(n^2+3) = n^3 + 3n^2 + 3n = (n+1)^3 - 1$. But two consecutive numbers cannot both be cubes unless they are $0$ and $1$. Since $n^2 + 3 \geqslant 3$, the pair $0$ and $1$ cannot arise in this way. Therefore the question is asking for something impossible (which I think is a bit sneaky! (Tongueout) ).[/sp]
[sp]Oops, yes, of course it should be $(n+3)(n^2+3) = n^3 + 3n^2 + 3n + 9 = (n+1)^3 + 8$. But the only cubes that differ by $8$ are $0$ and $\pm8$. Since none of those numbers is of the form $n^2+3$, no such $n$ exists.kaliprasad said:Sorry Opalg,
product you evaluated is not correct
Opalg said:[sp]
I still think it is unfair to phrase the question in a way that implies it is possible to find $n$, when in fact it is not possible.[/sp]
A perfect cube is a number that can be expressed as the cube of an integer. In other words, it is the result of multiplying a number by itself three times.
To find a number that satisfies both n+3 and n^2+3 being perfect cubes, we can use trial and error or algebraic methods. Trial and error involves testing different values for n until we find one that works. Algebraic methods involve setting up and solving equations with n as the variable.
There are restrictions for n in this problem. We are looking for positive integers that satisfy the conditions. Additionally, n must be a multiple of 3 in order for both n+3 and n^2+3 to be perfect cubes.
Yes, there can be multiple solutions for n. For example, if n=3, then both n+3 and n^2+3 are perfect cubes. However, there can also be cases where there is only one solution or no solution at all.
Problems like this can be used to exercise critical thinking and problem-solving skills. They can also be applied in fields such as cryptography, where finding specific types of numbers with certain properties is important.