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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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}&\...
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### 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 ...
261 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 $A^-_p$ ...
Let $p$ be an odd prime, and let $A_p$ denote the matrix $$[a_{ij}]_{1\le i,j\le (p-1)/2},$$ where $$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&... 3answers 880 views ### Is -\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2} always a square for each prime p\equiv 3\pmod 4? Let p be an odd prime and let S_p denote the determinant$$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$with (\frac{\cdot}p) the Legendre symbol. By Theorem 1.2 of my ... 2answers 362 views ### Does n^2 divide \det\left[\left(\frac{i^2+2ij+3j^2}n\right)\right]_{1\le i,j\le n-1} for each odd integer n>3? For any odd integer n>1 and integers c and d, define$$(c,d)_n:=\det\left[\left(\frac{i^2+cij+dj^2}n\right)\right]_{1\le i,j\le n-1},$$where (\frac{\cdot}n) is the Jacobi symbol. It is ... 0answers 121 views ### 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?... 0answers 246 views ### Expressing quartic Dirichlet characters modulo primes p\equiv 1\bmod 4 with Legendre symbols Looking through some old notes of mine from two years ago I found some crude notes writing what amounted to the statement that for any prime p\equiv 1\bmod 4 one could express for any odd integer p\... 0answers 219 views ### some phenomena about Fibonacci number [closed] Edit: for conjecture 3, define :(\frac{5}{F_n}) = 5 ^{\frac{F_n-1}2} mod (F_n), F_n  is odd number. When study Fibonacci number, some phenomena are observed： (\frac{F_p}{5})= (\frac{p}{5}) ... 2answers 268 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 ... 0answers 212 views ### 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 ... 1answer 2k views ### 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 ... 2answers 302 views ### overlap quadratic residues 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 ... 1answer 261 views ### How does this sequence grow 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-... 2answers 259 views ### Orders of the conjugates of an algebraic prime number in its residue field 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 ... 3answers 448 views ### Quadratic residues and nonresidues of arbitrary patterns Let p_1, p_2, \dotsc, p_n be distinct primes, and let \epsilon_1, \epsilon_2, \dotsc, \epsilon_n be an arbitrary sequence of 1 and -1. There is an integer a such that \left( \frac{a}{p_1} \... 2answers 245 views ### Impossible Range for Minkowski-Like Sum of Squares Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of Mx^2-N(a^2+b^2+c^2), is there any such z which is "deceptive", meaning that it ... 4answers 4k views ### Does there exist a non-square number which is the quadratic residue of every prime? I want to know whether there exist a non-square number n which is the quadratic residue of every prime. I know it is very elementary, and I think those kind of number are not exist, but I don't know ... 1answer 539 views ### Are there Carlitz analogues of quadratic residues and reciprocity? Let q be a prime power. I will use the notations of Keith Conrad's Carlitz extensions paper (but I'll work over \mathbb{F}_q rather than \mathbb{F}_p). The most general question I'm asking here ... 2answers 651 views ### quadratic residues - is there an easy explanation for the pattern I'm seeing? Let m be an integer and q be an odd prime factor of m^2 + 1. Is there an obvious reason that \left(\frac{2m}{q}\right) always equals 1? From some numerics, this seems to be the case. The ... 2answers 1k views ### Three consecutive quadratic residues problem Prove that doesn't exist N\in\mathbb{N} with property: for all primes p>N exist n\in\{3, 4,\ldots, N\} such that n, n-1, n-2 are quadratic residues modulo p. 0answers 304 views ### Shortest interval over which there are more quadratic residues than nonresidues Hi, I refer to formula (8) in Chapter 1 of H. Davenport, Multiplicative Number Theory, Third Edition, Springer (2000), which says that for primes q\equiv 3 \bmod 4:$$ L\left(\left(\frac{\cdot}{q}\...
Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$. Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the standard Gauss sum for \$...