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Do I need to use some kind of formula or just plug in numbers and find the one that works best?

What I tried doing so far is plugging in random numbers but whenever I multiply the four consecutive integers they always give me a square so adding $2$ makes it a non square root number (i.e.: $1 * 2 * 3 * 4 = 24 + 2 = 26$. $26$ does not have a square root.).

dxiv
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Daniel Marksman
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  • Not sure why you were adding $2$ there. In fact, we are not just talking about the least positive integer but the only integer that can be added to such a product and always give a square number. – Joffan Aug 10 '17 at 01:13
  • @DanielMarksman Please do not vandalize posted (and answered) questions, including your own. – dxiv Aug 15 '17 at 23:13

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Suggestive hint too long for a comment:

$$ 1 \times 2 \times 3 \times 4 + 1 = 25 = 5^2 $$ $$ 2 \times 3 \times 4 \times 5 + 1 = 121 = 11^2 $$ $$ 3 \times 4 \times 5 \times 6 + 1 = 361 = 19^2 $$

Ethan Bolker
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  • When you add one shouldn't the integer have to be positive which would make it 12? So 1 x 2 x 3 x 4 + 12 = 36 = 6^2? – Daniel Marksman Aug 14 '17 at 05:04
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    @DanielMarksman I don't understand your comment. $1$ is a positive integer. Yes, you can add $12$ to that product and get a square, but you can't add $12$ to every one of the products and get a square every time. – Ethan Bolker Aug 14 '17 at 11:40