Let $p$ be a prime and let $V(p)$ denote the Vandermonde determinant $$ \begin{array}{rcl} V(p) & = & \left| \begin{matrix} 1&1&1&\cdots&1&1\\ 0&1&2&\cdots&p-2&p-1\\ 0&1&4&\cdots&(p-2)^2&(p-1)^2\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\\ 0&1&2^{p-2}&\cdots&(p-2)^{p-2}&(p-1)^{p-2}\\ 0&1&2^{p-1}&\cdots&(p-2)^{p-1}&(p-1)^{p-1} \end{matrix} \right| \\ & = & \left| \begin{matrix} 1&1&1&\cdots&1&1\\ 0&1&2&\cdots&-2&-1\\ 0&1&4&\cdots&4&1\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\\ 0&1&2^{p-2}&\cdots&(-2)^{p-2}&-1\\ 0&1&2^{p-1}&\cdots&(-2)^{p-1}&1 \end{matrix} \right| \end{array} $$ of the elments of $\mathbb{Z}/p\mathbb{Z}$ ordered as $\{0,1,2,\dots,p-2,p-1\}$.
Computations suggest that
- when $p\equiv 1[4]$, then $V(p)$ is a root of $-1$,
- while if $p\equiv 3[4]$, $V(p)$ is most often $=1$, and sometimes $=-1$.
The first few values for which $V(p)=-1$ are:
[3, 7, 47, 59, 79, 83, 103, 107, 127, 139, 191, 199, 211, 251, 263, 283, 307, 331, 367, 379, 383, 431, 467, 479, 499, 503, 547, 587, 599, 607, 631, 643, 659, 727, 743, 811, 823, 827, 839, 859, 863, 883, 887, 907, 971, 991, 1087, 1151, 1163, 1171, 1259, 1283, 1307, 1319, 1367, 1423, 1427, 1459, 1483, 1487, 1511, 1523, 1531, 1583, 1619, 1627, 1663, 1699, 1783, 1787, 1811, 1823, 1871, 1951, 1979, 1987, 2011, 2027, 2039, 2063, 2083, 2099, 2111, 2143, 2179, 2243, 2251, 2267, 2287, 2311, 2339, 2383, 2411, 2423, 2447, 2459, 2467, 2503, 2539, 2551, 2591, 2699, 2707, 2711, 2719, 2731, 2767, 2843, 2851, 2879, 2887, 2971, 2999, 3019, 3067, 3079, 3119, 3167, 3187, 3191, 3203, 3251, 3299, 3319, 3331, 3347, 3359, 3391, 3407, 3467, 3491, 3499, 3511, 3527, 3539, 3559, 3571, 3583, 3623, 3671, 3739, 3779, 3803, 3823, 3863, 3907, 3923, 3931, 3967, 4007, 4019, 4051, 4079, 4099, 4127, 4139, 4211, 4219, 4259, 4271, 4423, 4447, 4519, 4567, 4583, 4591, 4603, 4663, 4831, 4931, 4999, 5003, 5023, 5059, 5087, 5099, 5107, 5147, 5167, 5171, 5179, 5227, 5323, 5351, 5387, 5431, 5503, 5507, 5519, 5531, 5563, 5623, 5647, 5651, 5659, 5683, 5711, 5743, 5783, 5791, 5807, 5827, 5839, 5879, 5903, 5923, 5939, 5987, 6011, 6067, 6079, 6091, 6131, 6143, 6163, 6211, 6263, 6299, 6311, 6343, 6359, 6367, 6491, 6547, 6551, 6563, 6571, 6599, 6607, 6659, 6719, 6779, 6791, 6803, 6823, 6863, 6871, 6899, 6967, 6983, 7019, 7027, 7043, 7079, 7103, 7151, 7159, 7207, 7211, 7219, 7351, 7451, 7459, 7487, 7507, 7523, 7547, 7591, 7603, 7607, 7687, 7691, 7699, 7703, 7727, 7759, 7867, 7879, 7919, 8039, 8087, 8111, 8167, 8219, 8287, 8291, 8311, 8363, 8419, 8443, 8467, 8543, 8699, 8707, 8719, 8779, 8783, 8807, 8831, 8839, 8863, 8867, 8923, 8971, 9043, 9059, 9103, 9127, 9203, 9283, 9311, 9319, 9403, 9419, 9463, 9479, 9511, 9539, 9551, 9587, 9619, 9631, 9643, 9719, 9743, 9767, 9787, 9871, 9907, 9931, 10007, 10039, 10091, 10139, 10151, 10159, 10163, 10211, 10243, 10247, 10259, 10267, 10271, 10331, 10391, 10427, 10459, 10463, 10487, 10531, 10559, 10651, 10663, 10667, 10711, 10723, 10799, 10867, 10883, 10891, 10903, 10939, 10979, 10987, 11003, 11027, 11047, 11071, 11083, 11087, 11279, 11287, 11299, 11311, 11351, 11399, 11411, 11491, 11519, 11527, 11551, 11579, 11719, 11779, 11807, 11839, 11903, 11939, 11971, 12071, 12143, 12163, 12203, 12239, 12251, 12263, 12343, 12347, 12491, 12503, 12539, 12583, 12647, 12739, 12743, 12763, 12799, 12911, 12923, 12967, 12983, 13003, 13043, 13099, 13127, 13147, 13151, 13163, 13183, 13187, 13219, 13259, 13267, 13327, 13399, 13411, 13487, 13523, 13567, 13591, 13691, 13711, 13759, 13807, 13831, 13859, 13879, 13903, 13963, 14107, 14159, 14207, 14243, 14303, 14323, 14423, 14431, 14447, 14503, 14627, 14683, 14699, 14723, 14731, 14747, 14759, 14767, 14771, 14783, 14827, 14831, 14843, 14851, 14867, 14879, 14887, 14891, 14939, 14951, 14983, 15083, 15091, 15107, 15131, 15187, 15199, 15287, 15307, 15331, 15383, 15391, 15451, 15511, 15551, 15559, 15643, 15667, 15727, 15739, 15767, 15803, 15823, 15887, 15907, 15923, 15971, 15991, 16007, 16063, 16103, 16139, 16267, 16319, 16363, 16411, 16447, 16451, 16487, 16547, 16567, 16603, 16651, 16703, 16747, 16759, 16763, 16787, 16811, 16831, 16843, 16927, 16979, 17011, 17123, 17167, 17183, 17203, 17207, 17239, 17299, 17359, 17387]
This sequence of numbers is known to the OEIS as A129518 and A260298 (but for the $2$)
I can prove the assertion relative to $V(p)$ being a root of $-1$ when $p\equiv 1[4]$ and $V(p)=\pm1$ otherwise: if $p\equiv 1[4]$, then on account of $\binom{p}2$ being even, we get that $$V(p)^2=\prod_{i\neq j}(j-i)=(-1)^p=-1$$ so that $V(p)$ is a root of $-1$, while if $p\equiv3[4]$, $\binom{p}2$ is odd and we get $$V(p)^2=-\prod_{i\neq j}(j-i)=-(-1)^p=1$$ so that $V(p)=\pm1$.
For $p\equiv 3[4]$, is it known when $V(p)=1$ and when $V(p)=-1$?
I can try to generalize the question to finite fields, and this requires us to order the elements of a finite field $\mathbb{F}$. We can forget about $0$ and use a generator of $\mathbb{F}^\times$ as a yard stick:
More generally, if $\mathbb{F}$ is a finite field of size $q$ and $\xi$ is a primitive element (i.e. generator of $\mathbb{F}^\times$), is the value of the determinant $|\xi^{ij}|_{0\leq i,j<q-1}$ known?
I found a reference discussing this determinant: Wilson Theorems for Double-, Hyper-, Sub- and Super-Factorials establishes that $V(p)\equiv(p-1)!!\mod p$, where $n!!$ is the double factorial, the product of the positive integers $\leq n$ with the same parity as $n$. Their Theorem 3 describes the sign $V(p) = (-1)^\nu$ for $p\equiv 3[4]$ in terms of $\nu$, the number of quadratic nonresidues $2< qnr < \frac{p-1}2$. And Theorem 4 gives yet another description of $V(p)$ in full generality.