The sign and congruence conjecture for tangent permanents

Let $p$ be an odd prime, put $m=(p-1)/2$, and define

$$T_p=\sqrt p\,\operatorname{per}\left[\cot\left(\frac{\pi jk}{p}\right)\right]_{1\le j,k\le m}. \tag{1}$$

Sun's Theorem 1.7 proves that $T_p\in\mathbb Z$ and $T_p\equiv1\pmod p$. Conjecture 4.7(ii) further predicts

$$\left(\frac{-1}{p}\right)T_p<0. \tag{2}$$

Both possible congruence classes of $p$ contradict this prediction:

$$T_{29}=84{,}643{,}985{,}981{,}440>0, \qquad T_{43}=-7{,}815{,}543{,}081{,}584{,}471{,}417{,}421{,}824<0. \tag{3}$$

Indeed, $29\equiv1\pmod4$, whereas $43\equiv3\pmod4$, so both numbers in (3) have the opposite sign to (2).

Here is an entirely exact certificate requiring no numerical approximations. Fix a prime $q\equiv1\pmod{4p}$, choose an element $w\in\mathbb F_q^\times$ of order $4p$, and set

$$\zeta=w^4,\qquad \iota=w^p,\qquad G=\sum_{a=0}^{p-1}\zeta^{a^2},\qquad g=\begin{cases}G,&p\equiv1\pmod4,\\G/\iota,&p\equiv3\pmod4.\end{cases} \tag{4}$$

The quadratic Gauss-sum evaluation gives $g^2=p$; more precisely, $g$ is the image of the positive real square root appearing in (1) under the cyclotomic reduction. Since

$$\cot\left(\frac{\pi r}{p}\right) =i\,\frac{e^{2\pi i r/p}+1}{e^{2\pi i r/p}-1}, \tag{5}$$

the image of the cotangent matrix over $\mathbb F_q$ has entries

$$c_{jk}=\iota\,\frac{\zeta^{jk}+1}{\zeta^{jk}-1}. \tag{6}$$

For subsets $S\subseteq\{1,\ldots,m\}$, define recursively

$$D_{\varnothing}=1,\qquad D_S=\sum_{k\in S}c_{|S|,k}D_{S\setminus\{k\}}. \tag{7}$$

Induction on $|S|$ shows that $D_S$ sums exactly the injections from the first $|S|$ rows onto $S$. Therefore

$$T_p\equiv gD_{\{1,\ldots,m\}}\pmod q. \tag{8}$$

For $p=29$, formula (7) gives the following three exact residues:

$$\begin{array}{c|r|r} q&w&T_{29}\bmod q\\ \hline 1{,}000{,}002{,}361&725{,}178{,}391&786{,}139{,}317\\ 1{,}000{,}003{,}057&194{,}657{,}493&727{,}227{,}789\\ 1{,}000{,}004{,}449&514{,}956{,}177&609{,}404{,}733 \end{array} \tag{9}$$

These residues reconstruct $84{,}643{,}985{,}981{,}440$ modulo

$$Q_{29}=1{,}000{,}009{,}867{,}031{,}322{,}291{,}111{,}000{,}073. \tag{10}$$

To certify uniqueness, write $H_m=\sum_{a=1}^m1/a$. The chord bound for sine gives

$$\left|\cot\left(\frac{\pi r}{p}\right)\right| \le\frac{p}{2\min(r,p-r)}. \tag{11}$$

For every nonzero row index $j$, the values $\min(jk\bmod p,p-(jk\bmod p))$, as $1\le k\le m$, are a permutation of $1,\ldots,m$. Bounding the permanent by the product of its absolute row sums and using $\sqrt p<p$, we obtain

$$|T_p|<p\left(\frac p2H_m\right)^m. \tag{12}$$

At $p=29$, this yields

$$|T_{29}|<29\left(\frac{33{,}980{,}257}{720{,}720}\right)^{14} <7{,}777{,}485{,}176{,}976{,}639{,}097{,}830{,}653 <\frac{Q_{29}}2. \tag{13}$$

Consequently, the Chinese remainder theorem proves the first equality of (3) as an identity of integers.

The same argument for $p=43$ gives

$$\begin{array}{c|r} q&T_{43}\bmod q\\ \hline 1{,}000{,}000{,}433&580{,}932{,}215\\ 1{,}000{,}003{,}013&789{,}445{,}710\\ 1{,}000{,}003{,}357&893{,}159{,}544\\ 1{,}000{,}004{,}389&2{,}428{,}947\\ 1{,}000{,}008{,}173&547{,}386{,}028 \end{array} \tag{14}$$

Here $43H_{21}/2=810{,}896{,}279/10{,}346{,}336$, and (12) gives

$$|T_{43}|<257{,}763{,}709{,}733{,}568{,}910{,}737{,}647{,}180{,}249{,}968{,}428{,}462{,}101 <\frac12\prod_{q\text{ in }(14)}q. \tag{15}$$

Thus (14) uniquely determines the second integer in (3). Both values also satisfy the independently known congruence $T_p\equiv1\pmod p$. The tangent-permanent sign assertion and the odd-composite divisibility assertion in the other parts of Conjecture 4.7 are not addressed here.

Source. Z.-W. Sun, Arithmetic properties of some permanents, Theorem 1.7 and Conjecture 4.7. The previously reported counterexample to the adjacent Conjecture 4.6 concerns sine/cosecant permanents and does not concern the cotangent permanents in (1).