Two consecutive positive integers' product in the form of a polynomial

Question

Two consecutive positive integers have the product in the form of $$n^2+10n+3$$ where $n$ is a natural number. Find the maximum value of $n$.

I really have no idea here. Substituting the two consecutive numbers in $a(a+1)$ gives the following:

$$a(a+1)=a^2+a=n^2+10n+3$$

Thanks for your help.

Answer 1

Hint: $\;\;(n+4)(n+5) \,\lt\, n^2+10n+3 \lt (n+5)(n+6)\;$ for $\;n \gt 17\,$.

Answer 2

$a^2 + a - (n^2 + 10n + 3) = 0 \implies$

$~\displaystyle a = \frac{1}{2} \left[-1 \pm \sqrt{1 + 4(n^2 + 10n + 3)} \right] = \frac{1}{2} \left[-1 \pm \sqrt{4n^2 + 40n + 13} \right].$

Let $D = 4n^2 + 40n + 13.$

Since $a$ has to be an integer, $D$ must be an odd perfect square.

Therefore, there must exist an integer $k$ such that

$4n^2 + 40n + 13 = (2k + 1)^2.$

$4n^2 + 40n + 13$ can be rewritten as $(2n + 10)^2 - 87.$

Therefore, you have that $(2n + 10)^2 - (2k+1)^2 = 87.$

Therefore, $(2n + 10 - 2k - 1) \times (2n + 10 + 2k + 1) = 87.$

Therefore,

$$(2n - 2k + 9) \times (2n + 2k + 11) = 87.\tag1 $$

Since the only factors of $(87)$ are $(1)$ and $(87)$,
and since the difference between the two factors
in (1) above is $(4k+2)$

you must have that $(4k + 2) = 86 \implies k = 21.$

Therefore, $(2n - 42 + 9) \times (2n + 42 + 11) = 87.$

Therefore $(2n - 33) \times (2n + 53) = 87.$

This makes $2n = 34 \implies n = 17.$

Edit
Analysis hole:

You can have that $(-1) \times (-87) = (87)$.

So, you can have that $(4k + 2) = - 86 \implies k = - 22$.

However, regardless of whether $k = -22,$ or $k = 21$, you still have that $(2k + 1)^2 = 1849.$

So, you still end up with $(2n + 10)^2 - 87 = 1849 \implies (2n + 10) = 44.$

The alternate computation of $(2n + 10) = -44$ is excluded both because $(n)$ is required to be a natural number and (also) because it is desired to find the maximum possible value for $(n)$.

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Addendum
Well, this is embarrassing.
Who knew that $(3 \times 29) = 87?$

Exploring this, if you have $(4k + 2) = (29 - 3)$

then $k = 6$.

This means that $(2n + 10)^2 - 87 = (13)^2 = 169.$

Therefore, you can have that $(2n + 10)^2 = 256 = (16)^2.$

Therefore, you can have that $(2n + 10) = \pm 16.$

Since it is desired to find the maximum value of $n$, the original answer of $(17)$ luckily still stands.