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### Quadratic residue problem involving prime divisors of a polynomial

Let $n$ be a square-free natural number, and let $f\in\mathbb{Z}[x]$ be monic and irreducible of degree $\geq2$. I am trying to determine whether there always exists a prime $p$, $p\nmid n$, ...
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### Mod n, are all higher powers also lower powers?

Since there are only finitely many residues mod $n$, there is some function $H(n) \le n$ such that for all integers $n>1$, $r$, and $e>H(n)$, if $r$ is an $e$-th power mod $n$ then there is some ...
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### Set of all primes $p$ that split in $\mathbb{Q}\left(\sqrt{-k}\right)$

Let $k$ be a squarefree positive integer. We know that a prime $p$ splits in $K=\mathbb{Q}(\sqrt{-k})$ if and only if $-k$ is a quadratic residue mod $p$. My question is: can we explicitly determine ...
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In $\mathbb{F_p}$, $p$ prime what is the larget subset $S$ of quadratic residues with no pair of elements differing by 1? In this related question Seva gives an example: "...assuming $p\equiv\... • 4,792 2 votes 1 answer 513 views ### Quadratic non-residue problem For a positive integer$n$, let$a(n)$the smallest number$k>0$such that$-n$is not a quadratic residue modulo$k$. Using CRT, we can prove that all values of$a(n)$are prime powers, and every ... • 593 5 votes 0 answers 502 views ### Two conjectures for primes$p\equiv 1\pmod 8$Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes$p\equiv1\pmod8$based on my ... • 14.5k 1 vote 0 answers 83 views ### How to describe the automorphisms of$\mathbb{Q}(\theta)$that fix$\mathbb{Q}(\sqrt{-m})$Suppose$m > 0$is a square free integer and$m \equiv 1 $mod$4$. Then the$K = \mathbb{Q}(\sqrt{-m})$is a subfield of$L = \mathbb{Q}(\theta)$, where$\theta$is a primitive$4m$-th root of ... • 697 1 vote 1 answer 140 views ### Sum of an arithmetic sequence involving Euler factors I am trying to find an asymptotic formula for the following sum as$T \to \infty$. $$\sum_{t = 1}^{T} \prod_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)... • 577 0 votes 0 answers 97 views ### Primes in residue classes [duplicate] For which sets of residue classes are there easy elementary proofs that there are infinitely many primes in them, which don’t require the machinery of proofs of Dirichlet’s theorem? Example: it’s ... • 827 1 vote 0 answers 87 views ### Evaluate \det[[\lfloor\frac{aj-(a+1)k}n\rfloor]_q]_{1\le j,k\le n} and \det[[\lceil\frac{(a+1)j-ak}n\rceil]_q]_{1\le j,k\le n} The q-analogue of an integer m is defined by [m]_q=(1-q^m)/(1-q). Note that \lim_{q\to1}[m]_q=m. I have formulated the following conjecture on determinants involving the floor function and the ... • 14.5k 11 votes 0 answers 421 views ### effective and unconditional upper bound for the smallest quadratic residue Let p be a prime number, and let r=r(p) be the smallest prime number with (r/p)=1. The classical result of Linnik-Vinogradov (based on Burgess) implies that r\ll_\epsilon p^{1/4+\epsilon}, but ... • 1,130 4 votes 1 answer 227 views ### Counting squares modulo p that are also prime in an interval What would be the best lower bound for the number of squares modulo p in an interval [1,N] with N<p that are prime? Via the Burgess bound, I can find a lower bound for the number of squares ... • 41 3 votes 1 answer 406 views ### Does each prime p>541 have a quadratic residue x^4+y^4<p? For any prime p>5, one of the numbers$$1^2+1=2,\ \ 2^2+1=5,\ \ 3^2+1=10=2\times5$$is a quadratic residue modulo p. In 2014 I conjectured that each prime p has a primitive root g<p of ... • 14.5k 3 votes 1 answer 222 views ### Pythagorean triples and quadratic residues modulo primes QUESTION. Are my following conjectures true? How to prove them? Conjecture 1. For each prime p>100, there are a,b,c\in\{1,\ldots,p-1\} such that$$\left(\frac ap\right)=\left(\frac bp\right)=\... • 14.5k 4 votes 1 answer 252 views ### 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 ... • 1,666 12 votes 1 answer 738 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$\mathbb{Q}(\sqrt{-p})$... 4 votes 1 answer 218 views ### Distribution of quadratic residues in an interval For a prime (or prime power)$p$and some absolute constant$C$(say$C$= 100), consider the set$A$of all$1 \leq a \leq p/C$such that$1 \leq a^2 \leq p/C$modulo$p$. Is it known that$|A| = \...
I'm wondering if it's possible, given a prime p and an infinite list of primes $q_1$, $q_2$, ... to find an integer d which (1) is not a square mod p, but (2) is a square mod $q_i$ for all i. Always, ...