# Tag Info

• 12.4k
Accepted

### How does this sequence grow

The answer is yes, and the number of solutions with a prime $p$ is $\lfloor \frac{p+5}{8} \rfloor$ when $p \not\equiv 1 \pmod{8}$ and is $\lfloor \frac{p+5}{8} \rfloor + 1$ when $p \equiv 1 \pmod{8}$. ...
• 17.5k

### On $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i+j}(2i+j)$ and $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i-j}(2i-j)$ modulo a prime $p>3$

Here is a respectively short way to write down what we came up with Dmitry Krachun tonight. Denote $p=2m+1$. The idea is very simple: calculate the product \prod_{j\in\{s,s+1\}, 1\leqslant i\...
• 86.9k
Accepted

I don't think this is true in general, for instance, if there exists $1\leq i <j<k<r$ such that $n_in_j = n_k$, then $(n_i/p)=(n_j/p)=-1\implies (n_k/p)=1$ (where $(a/p)$ is the Legendre ...
• 1,076
Accepted

### Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, A Classical Introduction to Modern Number Theory). In fact, Theorem 5 of Chapter 8 (on page ...
• 4,320

### Counting squares modulo $p$ that are also prime in an interval

Here is the paper by P. Pollack on the distribution of non-residues and residues. Theorem 1.3 states that for any $\varepsilon>0$, $A<\infty$ and large enough $m$ there are at least $(\ln m)^A$ ...
• 5,084

### On triangular numbers modulo primes

Differences A general useful fact (compare with my answer to your previous question) is that whenever we have $A=\{a_1,\dots,a_n\}\subset \{0,1,\dots,p-1\}$ such that $n=|A|$ is odd, the sign of a ...
• 86.9k
Accepted

### Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

This is an NP-hard problem. That is, Manders and Adleman [1] proved that given $a$, $b$, and $c$, it is NP-complete to determine if there exists $x\le b$ such that $x^2\equiv a\pmod c$, and that it ...
• 38.8k

### A conjecture on primitive tenth roots of unity

Not a complete solution. Let $p$ be (1 mod 4) and $r,n$ runs over quadratic residues/non residues mod $p$ in $[1,p-1]$ and let $R_p(x)=\prod_r(x-\zeta_p^r),\;\;\; N_p(x)=\prod_n(x-\zeta_p^n).$ ...
• 457

### Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

Not exactly an answer, but in exercise $18$ in page $106$ of Ireland and Rosen's A Classical Introduction to Modern Number theory it states: Let $p\equiv 1\mod 4$ and let $p=A^2+B^2$ where we fix $A$ ...
• 51
Accepted

• 1,596
### Does $n^2$ divide $\det\left[\left(\frac{i^2+2ij+3j^2}n\right)\right]_{1\le i,j\le n-1}$ for each odd integer $n>3$?
This is not an answer, either. Just some attempts to attack the problem. Let $r(n)$ be the square-free part of $(2, 3)_n$, i.e. $r(n)$ is square-free and we have $(2, 3)_n = r(n) B^2$ for some ...
### Does $\det[\lfloor(i^2+j^2)/p\rfloor]_{1\le i,j\le(p-1)/2}$ vanish for each prime $p>7$ with $p\equiv3\pmod4$?
Let $p$ be large enough. Then there are two pairs of consecutive squares $a$, $a+1$ and $b$, $b+1$ modulo $p$ (otherwise the parities of sqiares modulo $p$ cannot alternate more than constant times, ...