# Tag Info

Accepted

### Show $x^p-x$ is not in $\ker\Psi_p$

It is obviously not in the kernel by looking at the coefficients of the polynomial. To prove that all evaluations are $0$, use Lagrange's theorem for the multiplicative group of the field of $p$ ...
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Accepted

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1 vote
Accepted

1 vote

### Factors of a quadratic in modulo $9$

Assume that $$f(x)\equiv (x-b)(x-c)\pmod9.$$ You already observed that $bc\equiv1\pmod9$, so also $bc\equiv1\pmod3$. It follows that $b$ and $c$ must congruent to each other modulo three. For ...
• 135k
1 vote

### How to prove that all elements inside a cycle of a cyclic group are different from each other

Here is the standard way to see this. Suppose that $n = ord(a)$, so $n$ is the smallest positive integer such that $a^n = e$. Then note the following: (*) $a^k = e$ if and only if $n$ divides $k$. ...
• 1,033

### Find All Solutions to System of Congruence

You did it well until the computation of $x$. You forgot to do the multiplication by $3$ in the last term: $$x=(-20*1)*2-(15*1)*1+(12*3)*3\equiv 53\pmod{60}$$
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• 275k
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Accepted

### Why is the order of an element equal to the order of the group it generates?

Suppose that $ord(a) = n$ for some integer $n > 0$. Then $a^n =e$. Thus $a \circ a^{n-1} = e$, and multiplying both sides with $a^{-1}$ gives you $$a^{-1} = a^{n-1}.$$ Now raising both powers by ...
• 1,033
1 vote

### Question involving Fermat's Little Theorem

What you are doing wrong is applying Fermat’s Little Theorem to $m=2p$, which is not prime. For composites, you should use Euler’s theorem.
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1 vote

### Question involving Fermat's Little Theorem

As mentioned in the comment, Fermat's little only applies to primes. Since $p$ and $2$ are coprime, it suffices to show $2\mid 2^{2p-1}-2$ (which is obvious as both of $2^{2p-1}$ and $2$ are even), ...
• 17.8k
1 vote

### If $d\mid10a-1$ then $d\mid10q+r\iff d\mid q+ar$

$d\mid10a-1\;.$ Prove that $\;d\mid10q+r\iff d\mid q+ar\;.$ Proof$\,\implies:$ Since $\;d\mid10a-1\,,\;$ it follows that $\;d\mid r(10a-1)\;.$ Since $\;d\mid r(10a-1)\;$ and $\;d\mid 10q+r\;,\;$ we ...
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### What is the order of $p_{1}^{x} \bmod{n}$ where $p_1$ is a prime factor of $n$

Hint: $\!\bmod n\,$ suppose $p^{\Bbb N}$ has preperiod $j$ and period $k,\,$ i.e. $j,k$ are minimal such that $\,p^{j+k}\equiv p^j.\,$ Then $\,n\mid p^j(p^k-1)\,$ so $\,p^j\mid\mid n\,$ and $\,k\,$ ...
• 275k
1 vote

• 16.4k
### Is there a way to show that the Fibonacci subsequence $F_{6n+2}+2$ can't have any square number?
Render the Fibonacci Sequence $\bmod 4$: $1,1,2,3,1,0,1,1,2,...$ The sequence has period $6$ and $F_{6n+2}\equiv1\bmod4$. Therefore $F_{6n+2}+2\equiv3\bmod4$ and ... .