An integer $a$ is a quadratic residue modulo $p$ if $a \equiv x^2 \pmod p$ for some integer $x$. If $a$ is not a quadratic residue modulo $p$, it is said to be a quadratic nonresidue modulo $p$.

811 questions
Filter by
Sorted by
Tagged with
384 views

### Some congruence problem

In my work, I reached the following congruence. Here, $\square_1$, $\square_2$ are square numbers and $p$ denotes the prime number. \begin{align} &4+1\equiv\square_1\text{ modulo }4p, \\&4(...
• 373
62 views

• 3,940
47 views

• 511
49 views

### How to find all $p$ congruences such that $(\frac{-5}{p})$ are quadratic residues

I am self-studying number theory and came across the question asked by another user in this post here. Specifically part (b). My initial solution is to use the fact that the Legendre Symbol is ...
29 views

### Do we need relative prime in the definition of a quadratic residue? [duplicate]

From my lectures I know that $a$ is a quadratic residue mod $m$ if there is a $b$ such that $b^2\equiv a \pmod m$. But in several book I now read that $a$ and $m$ have to be coprime. That would mean ...
• 434
131 views

### Prove that if $p \equiv 7 \pmod{8}$, the solutions to the congruence $x^2-34x+1 \equiv 0 \pmod{p}$ are both quadratic residues modulo p.

Consider the quadratic congruence $$x^2-34x+1 \equiv 0 \pmod{p},$$ where $p \equiv 7 \pmod{8}$. There are two solutions, since we can write it as (x-17)^2 \...
1 vote
32 views

• 434
51 views

### How to show a quadratic polynomial with complex roots has solutions mod $p$?

When the discriminant is $-k$ or ${-1\over k}$: As long as $-1$ and $k$ are both squares $\bmod p$ (trivial) or are both not squares, there are solutions mod $p$. How can this be explained/proven? e.g....
152 views

### Is there any solution to $x^2 \equiv 53 \mod97$

Is there any solution to $x^2 \equiv 53 \mod97$ ( or in other words: is 53 quadratic residue of 97 ). It is possible to place any of $0,1,...,96$ in the congruence and check if any of them is a ...
• 171
75 views

### A Diophantine equation centered around the divisor counting function and squares

For each positive integer $n$, let $τ(n)$ be the prime counting function. Prove that for all positive integers $a$ and $b$ satisfy the following equation that $a+b$ is even: $a + τ (a) = b^2 + 2$. So ...
112 views

### Principal square root of quadratic residue mod $n$, where $n$ is a Blum integer

Let $n = pq$ be a Blum integer, where $p$ and $q$ are prime numbers, and $y$ belongs to the Quadratic Residues modulo $n$ ($y \in QR(n)$). How could I prove that the primitive square root $x$ of $y$ ...
100 views

For my thesis I need a good bound for the least quadratic non-residue modulo an odd prime $p$, which I can cite as proven. So I researched a lot and found several papers and bound. As far as I read in ...
• 434
99 views

### Experimentally found weird number theory pattern $p^\frac{d-3}{2}\cdot \left(p^\frac{d-1}2 \pm \{-1 \text{ or } p-1 \}\right) \mod d$ equals $0,1$

In the course of studying a problem involving counting points on hyperspheres and hyperplanes in dimension $d$ mod $p$, I came across the following interesting pattern Conjecture/Question: for (odd?) ...
• 8,945