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# A perfect square

Prove that for any natural number $$x$$, $$x(x+1)(x+2)(x+3)+1$$ is a perfect square.

Note by Akarsh Jain
5 months, 1 week ago

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I don't think this a legit proof but here it is.

the expression can be written as

$$\large (x^{2}+3x)(x^{2}+3x+2)$$ $$\text{Now Substitute} \hspace{1mm}(\large x^{2}+3x) =y$$.

The above expression changes to $$\large (y)(y+2)$$ = $$\large y^{2}+2y$$

On Adding 1 gives us $$\large y^{2}+2y+1$$= $$(y+1)^{2}$$

$$\large QED$$ · 5 months, 1 week ago

That works!

Note that you have to show that y+1 is an integer, so that the number is a perfect square (as opposed to a perfect square as an expression). Staff · 5 months, 1 week ago

Thanks

it is mentioned that $$x$$ is a natural number $$\therefore$$ $$x^{2}+3x$$ must also be an integer. This proves that $$y$$ must also be an integer.

Does this complete the proof? · 5 months, 1 week ago

Yes.

(likely just a typo) Note that you want $$x^2 + 3x = y$$. Staff · 5 months, 1 week ago