# Questions tagged [legendre-symbol]

For questions involving the Legendre symbol, $\genfrac{(}{)}{}{}{a}{p}$ for integer $a$ and prime $p$.

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### Can at least one primitive root $w$ of $N$ be expressed as $a^2-b$, where $(b|N)=-1$

I am stuck on a thought experiment: can any (or for that matter, at least one) primitive root $w$ of $N$, $N$ prime, be expressed as $w=a^2-b$, where $(b|N)=-1$, and $a,b\in\mathbb{N}$. We know that ...
35 views

### Assume that p and q are odd primes and $p \equiv q \pmod {28}$. Show that $(\frac{7}{p}) = (\frac{7}{q})$.

Assume that p and q are odd primes and $p \equiv q \pmod {28}$. Show that $(\frac{7}{p}) = (\frac{7}{q})$. Could anyone give me a hint for the solution please?
55 views

### Why if the Legendre symbol satisfy $\left(\frac{a}p\right)=\left(\frac{p}a\right)$ then $\left(\frac{a}p\right) = 1$?

Sorry for my stupid question: This is in completion to this question Let $p$ be a prime of the form $p = a^2 + b^2$ with $a,b \in \mathbb{Z}$ and $a$ an odd prime. Prove that $(a/p) =1$ Why if the ...
34 views

### Exponentiation with odd number in modular arithmetic

Show that if $n\geq3$ is odd, then $2^n-1\equiv7\mod24$. I tried solving this backwardly. We want to prove that $2^3(2^{n-3}-1)=2^n-2^3\equiv0\mod24$. Since $\frac{24}{2^3}=3$, this leaves us to ...
39 views

### Question on modular arithmetic with prime numbers

Let $p\neq3$ be a prime conguent to $3\pmod4$ and let $q$ be a prime divisor of $(12p)^{2019}+1$ satisfying $q\equiv p^2+1\pmod{3p}$. Determine $q\pmod 4$. I tried solving the problem as follows. ...
44 views

### Proving that $n$ is not a prime [duplicate]

Let $n=3^{100}+2$ and assume that $X^2-53$ does not have zeroes in $\mathbb{Z}/ n\mathbb{Z}$. Show that $n$ is not a prime. I tried solving this problem by assuming that $n$ is a prime (in order to ...
40 views

### Modular arithmetic with Legendre symbol

Let $n\in\mathbb{Z}_{>0}$ and let $p\neq3$ be a prime divisor of $n^2+n+1$. Show that $p\equiv1\mod3$. I thought of trying to prove that $\left(\frac{p}{3}\right)=1$, since 1 is the only element ...
79 views

### establish that every prime number p of the form $8k + 1$ or $8k +3$ can be written as $p = a^2 + 2b^2$ for some choice of integers a and b.

The question is: Establish that every prime number p of the form $8k + 1$ or $8k +3$ can be written as $p = a^2 + 2b^2$ for some choice of integers a and b. And the Hint says: Mimic the proof of ...
27 views

### Is the Legendre symbol (9/8) the same as Legendre symbol $3^2/2$? [closed]

Is the Legendre symbol (9/8) the same as Legendre symbol $3^2/2$? if so, by what property?
69 views

### How can I prove that $(3/p) = -1$ if $p \equiv \pm 5 \pmod {12}$

I know how to prove that $(3/p) = 1$ if $p \equiv \pm 1 \pmod {12}$ but I need to prove that $(3/p) = -1$ if $p \equiv \pm 5 \pmod {12}$, which the book write it as it is without explaining why ...