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### effective and unconditional upper bound for the smallest quadratic residue

Let $p$ be a prime number, and let $r=r(p)$ be the smallest prime number with $(r/p)=1$. The classical result of Linnik-Vinogradov (based on Burgess) implies that $r\ll_\epsilon p^{1/4+\epsilon}$, but ...
166 views

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

What would be the best lower bound for the number of squares modulo $p$ in an interval $[1,N]$ with $N<p$ that are prime? Via the Burgess bound, I can find a lower bound for the number of squares ...
325 views

### Does each prime $p>541$ have a quadratic residue $x^4+y^4<p$?

For any prime $p>5$, one of the numbers $$1^2+1=2,\ \ 2^2+1=5,\ \ 3^2+1=10=2\times5$$ is a quadratic residue modulo $p$. In 2014 I conjectured that each prime $p$ has a primitive root $g<p$ of ...
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### Dominating sets in subtournaments of the Paley tournament

For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices. Is ...
485 views

### On sums of quadratic residues

Let $p>3$ be a prime. We set $R=\{x\in\mathbb{Z}: (x/p)=1\}$, where $(\cdot/p)$ is the Legendre symbol. When $p\equiv3\pmod4$, by class formulae of imaginary quadratic fields $\mathbb{Q}(\sqrt{-p})$...
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### Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

As the title asks: does there exist $N$ such that, for any prime $p$ larger than $N$, the expression $x^4 +y^4$ takes on all values in $\mathbb{Z}/p\mathbb{Z}$? I have been thinking about this ...
73 views

### quadratic residues and cubic polynomials [closed]

I'm really not sure about this, but I've heard somewhere that for any prime $p$, $|\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} ) |\le \sqrt{2p}$ holds. Does anyone know a proof for this inequality ...
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### applications of finding least quadratic nonresidue mod $p$?

I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$. My question is that why it is so ...
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### A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?

I have the following conjecture involving a possible new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$ with $p$ an odd prime. Conjecture. Let $p$ be an odd ...
113 views

### Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?

Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define $$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$ where $(\frac{\cdot}p)$ is the Legendre ...
198 views

### On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$

Let $p$ be an odd prime. For $d\in\mathbb Z$ we define $$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$ where $(\frac{\cdot}p)$ is the Legendre symbol. By (1.17) of my ...
311 views

### Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by $$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$ In 2004, R. Chapman [Acta ...
391 views

### A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)

As in Question 319254, for an odd prime $p$ and integers $c,d$ we define $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right),$$ where $(\frac{\cdot}p)$ is the Legendre symbol. In my ...
108 views

### A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (II)

As in Question 319254, for an odd prime $p$ and integers $c,d$ we let $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right).$$ If $p\equiv1\pmod4$, then we obviously have \begin{align}&\...
151 views

### A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (I)

Let $p$ be an odd prime. Here I introduce the sum $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right)$$ with $c,d\in\mathbb Z$, where $(\frac{\cdot}p)$ is the Legendre symbol. I have a ...
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### Permutations of squares and finite fields

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let $$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$ Motivated by Question 316142 of mine, here I ask the following ...
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### 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 $\lfloor x\rfloor$ be the floor function. QUESTION: Does the determinant $$D_p=\det\left[\left\lfloor\frac{i^2+j^2}p\right\rfloor\right]_{1\le i,j\le(p-1)/2}$$ vanish for each prime $p>7$ with ...
306 views

### On triangular numbers modulo primes

Let $p$ be an odd prime. For $a\in\mathbb Z$ let $\{a\}_p$ denote the least nonnegative residue of $a$ modulo $p$. The list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ is a permutation of all the quadratic ...
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### A new determinant question for primes $p\equiv3\pmod4$

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by my question http://mathoverflow.net/questions/310301, here I introduce the matrices $A^+_p$ and $A^-_p$ ...
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### Does each prime $p>3$ have a quadratic nonresidue which is a Mersenne number?

Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime. QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?...
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359 views

### 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)?

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)? Using the Tonelli-Shanks algorithm and the Chinese remainder ...
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### Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues". I recently ...
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### Relationship between quadratic residues modulo a prime and quadratic residues modulo a prime power [closed]

Been going through Alan Baker's A Comprehensive Course in Number Theory. Very interesting book, although the way proofs are presented sometimes throws me off a little. I usually read through a ...
312 views

Let $p$ be a prime number of form $4k+1$ and $M$ is its quadratic residue set. Let $M_i=\{i+x|\forall x\in M\}$ $\forall 0<i<p$. Does there exist a positive constant $\varepsilon$ such that ...
Let $a(n)$ be the number of solutions of the equation $a^2+b^2\equiv -1 \pmod {p_n}$, where $p_n$ is the n-th prime and $0\le a \le b \le \frac{p_n-1}2$. Is the sequence $a(1),a(2),a(3),\dots$ non-...
Of interest to me is the following question (it would be nice to find out what is known in its direction): Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime ...