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When $n=4,4\cdot n!-4n+1=4\times24-4\times4+1=9^2.$

I wonder if $4\cdot n!-4n+1$ can be a perfect square when $n>4$?

I searched $n$ from $5$ to $10000$ but no qualified number was found.

I know that:

(1) $n$ is even, since $-4n+1 \equiv1 \mod 8$;

(2) if $p|4n-1$ and $p\leq n/2$ then $p^2|4n-1$, here $p$ is prime;

(3) if $p \not|4n-1$ and $p\leq n$ then $\left (\frac{-4n+1}{p}\right) = 1$, here $p$ is prime, $\left (\frac{a}{p}\right)$ is Jacobi Symbol.

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    $\begingroup$ This is equivalent to $n((n-1)!-1)=m(m+1)$, maybe this helps (or maybe not). $\endgroup$ – YiFan Oct 12 at 11:52
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    $\begingroup$ I think it's still not known whether $n!+1$ can be a square for $n$ other than $4,5,7$. See oeis.org/A085692 – so there can be very hard problems not a million miles removed from yours. $\endgroup$ – Gerry Myerson Oct 12 at 12:12
  • $\begingroup$ $(n-1)!-1$ doesn't have smaller divisor than $n$. $\endgroup$ – Takahiro Waki Oct 12 at 12:26
  • $\begingroup$ @GerryMyerson As far as I know this has been solved in the meantime, but how is that related to this question ? $\endgroup$ – Peter Oct 12 at 12:46
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    $\begingroup$ checked up to $10^9$: for(n=1, 10^9, a= 1-4*n; t= 1; forprime(p=2, n, if(kronecker(a,p)==-1, t= 0; break()); ); if(t, b= 4*n!+a; if(issquare(b), print(n" "a" "b))); ) $\endgroup$ – Dmitry Ezhov Oct 13 at 1:21
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If $\nu_p(4n-1)< \nu_p(n!)$ for all primes $p \mid (4n-1)$, then we have- $$x^2 = 4n!-4n+1=(4n-1)(\frac{4n!}{4n-1}-1)$$ and for every $p \mid (4n-1)$, we also have $p \mid \frac{4n!}{4n-1}$. Thus- $$\gcd\bigg(4n-1,\frac{4n!}{4n-1}-1\bigg)=1 \implies (4n-1)=x_1^2$$ However, this is impossible as $4n-1 \equiv 3 \pmod{4}$ and $3$ is not a quadratic residue modulo $4$.

Next, we try to find all values of $n$ for which $\nu_p(4n-1) \geqslant \nu_p(n!)$ for some odd prime $p \mid (4n-1)$. Let $\nu_p(4n-1)=t$ and $4n-1=kp^t$ for positive integers $k$ and $t$. We have- $$4n-1=kp^t \implies n=\frac{kp^t+1}{4}\geqslant p\bigg\lfloor\frac{kp^t+1}{4p}\bigg\rfloor$$

Thus, we use the first the first term in Legendre's formula to establish- $$\nu_p(n!) \geqslant\bigg\lfloor\frac{kp^{t}+1}{4p}\bigg\rfloor$$

We must then have- $$t\geqslant\bigg\lfloor\frac{kp^{t}+1}{4p}\bigg\rfloor$$

Using some grunt work checking, it is quite easy to see that the only cases for $t>1$ are: $$(t,k,p)=(2,1,3),(2,1,5),(2,1,7),(2,1,11),(2,2,3),(2,2,5),(2,3,3),(3,1,3)$$ and we have $4n-1=kp^t$, which shows that- $$4n-1 \in \{9,25,49,121,18,50,27,27\} \implies 4n-1\in \{9,18,25,27,49,50,121\}$$

As $4n-1 \equiv 3 \pmod{4}$, the only possibility is $4n-1=27$ showing $n=7$, which fails.

For $t=1$, we get $1 \geqslant \big\lfloor\frac{k+1}{4}\big\rfloor$. Thus, we must have $k<7$. As $4n-1$ is odd, so is $k$. Thus, we have the only possibilities being- $$4n-1 \in \{p,3p,5p,7p\} \implies n \in \bigg\{\frac{p+1}{4},\frac{3p+1}{4},\frac{5p+1}{4},\frac{7p+1}{4}\bigg\}$$

for some prime $p$.

Now, if $4n-1=qp$ where $q \in \{3,5,7\}$, since $t=1$, we have $q \neq p$. Then, we will have $$\nu_q(n!) \leqslant \nu_q(4n-1)$$ since otherwise, the expression will have an odd power of $q$ which is prime in all cases. It can easily be seen that this means $p<8$ which gives us the work of checking- $$4n-1 \in \{9,15,21,25,35,49\}$$ out of which the only values $3 \pmod{4}$ are $15$ and $31$, which give the cases $n=4,8$. Only $n=4$ works.

Finally, we are left with the case $4n-1=p$ i.e. $n=\frac{p+1}{4}$ where $p \equiv 3 \pmod{4}$ is a prime. We have- $$x^2=4\bigg(\frac{p+1}{4}\bigg)!-p$$

Consider a prime $q \leqslant \frac{p+1}{4}$ (Not as defined earlier). We have- $$\bigg(\frac{p}{q}\bigg)\bigg(\frac{q}{p}\bigg)=(-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}}$$

Since $x^2 \equiv -p \pmod{q}$ must be a quadratic residue, we have- $$\bigg(\frac{-p}{q}\bigg) = 1 \implies \bigg(\frac{p}{q}\bigg)=(-1)^{\frac{q-1}{2}} \implies \bigg(\frac{q}{p}\bigg)=(-1)^{\frac{p-3}{2}\cdot\frac{q-1}{2}}=1$$ as $p \equiv 3 \pmod{4}$.

This means that all odd primes till $\frac{p+1}{4}$ are quadratic residues modulo $p$. One can check the cases below to show $p > 7$ which shows that- $$4\bigg(\frac{p+1}{4}\bigg)!\equiv 0 \pmod{8} \implies x^2 \equiv -p \equiv 1 \pmod{8} \implies p \equiv 7 \pmod{8}$$

This shows that $2$ is also a quadratic residue modulo $p$. Since all primes till $\frac{p+1}{4}$ are quadratic residues, all numbers till $\frac{p+1}{4}$ are quadratic residues too.

This means that we already have $\frac{p-3}{4}$ cases where a quadratic residue is followed by another quadratic residue. However, for $p \equiv 3 \pmod{4}$, this must occur exactly $\frac{p-3}{4}$ times. This means that for the following cases, we will never have consecutive quadratic residues.

This can be proved incorrect with the first value succeeding $\frac{p+1}{4}$ that is $a \equiv 2 \pmod{6}$ as $a$ and $a+1$ have all prime factors less than $\frac{p+1}{4}$.

Thus, we have the only solution being $n=4$ in the positive integers.

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  • $\begingroup$ First sentence, why is it that prime $p$ dividing $4n-1$ implies $p$ divides $4n!/(4n-1)$? $\endgroup$ – Gerry Myerson Oct 13 at 11:48
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    $\begingroup$ As $\nu_p(4n-1)<\nu_p(n!)$ $\endgroup$ – Haran Oct 13 at 12:02

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