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Nov
10
comment Numbers $n$ whose prime factors are $2$ and $5$ if and only if $\sum_{k=1}^{rad(n)}\mu(k)k=0$?
For $n$ small n=10 is the only solution to $\sum_{k=1}^n \mu(k)k=0$. This means that $\sum_{k=1}^{\operatorname{rad}(n)} \mu(k)k=0$, for all the members of the sequence A033846 as they all have $\operatorname{rad}(n) = 10$. The term $\sum_{k=1}^{\operatorname{rad}(n)} k=0$ seems to be an error.
Sep
24
comment Factorization in a Group
@kyloc, I found the paper and it directly cover your case so I'm updating the answer.
Sep
21
comment The most Efficient Algorithm for Factoring Polynomial Over Finite Field
Why not? All the residues will be 256-bit numbers and all the polynomials will have less than $n$ terms, with $n$ the degree of $F$. I'm not sure if you can hope for anything faster.
Jun
15
comment Is there any polynomial function $f$ such that If $\gcd(p,q)=1$ then $\gcd(f(p),f(q))=1$ for all such $p,q$?
How about f(x) = x?
May
5
comment Using gauss's lemma to find $(\frac{n}{p})$ (Legendre Symbol)
Be careful, you have an error in the set: it is not $\{3,6,\dots,3\frac{p-1}{a}\}$ but $\{3,6,\dots,3\frac{p-1}{2}\}$ in this case it doesn't alter the result but it would for $a=5$ or bigger.
Mar
3
comment Is $53$ expressible in this form?
Ark I used PARI GP, the powers of 3 mod 117 are just 3,9,27,81 and the powers of 2 are 1, 2, 4, 8, 11, 16, 22, 32, 44, 59, 64, 88 so it is easy to check
Oct
21
comment Proof of $p_n<n^2$ by Elementary Means
The statement is not true for $p_1=2$, so you need an exception for that case.
Sep
26
comment Infinitely many primes congruent to 1 mod prime
You are right the proof is incomplete, let me think if I can find a way to patch it otherwise I will remove it.
Sep
5
comment Number of pairs $(i, j)$ where $1\leq i < j \leq N$ such that $i|j$
Thanks for your comments, I've edited the answer with your sugestions.
May
12
comment Euler's Refutation of Fermat's Conjecture
Fermat statement was about the integers $2^{2^n}+1$.
Mar
31
comment quadratic residues and prime divisor
As far as I know, we don't know if there are infinitely many integers $n$ such that $n^2+1$ prime, it is an open problem.
Mar
17
comment Properties of a certain integer sequence
Carrying the computation a little further gives the sequence: 1, 1, 2, 2, 3, 5, 6, 8, 11, 17, 25, 33, 41, 52, 80, 139, 204, 245, 289, 410, 692, 1159, 1477, 2010, 2769, 4247, 6128, 7709, 9817
Mar
15
comment Reducing simultaneously a pair of fractions $\frac{a^2}{b},\frac{ a^3}{c}$ using only gcds
You are right, there is no known algorithm even to know if an integer is square-free which does not rely in factorization. Thanks, I hadn't think in your example.
Mar
14
comment Reducing simultaneously a pair of fractions $\frac{a^2}{b},\frac{ a^3}{c}$ using only gcds
You would need either to factor $g$ in order to find $d$ or to iterate over the integers up to $\sqrt[6]g$. However I'm changing the question to allow root extraction and avoid iteration. What I'm looking for is a fast way to compute it.
Mar
1
comment Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$
Could you give a reference to understand how the value $g_3 = \eta^6/16$ is found? Thanks.
Jun
4
comment Can this sum be simplified to closed form?
I'm not sure how to prove it but it seems that the following might be true: set $S(L,b) = \sum_{a=0}^{L-1} \lfloor ab/L\rfloor^2$, and write $L = nb+t$ then $S(L,t) = n(n+1)(2n+1)/6 + S(t,b)$.
May
4
comment Do these inequalities regarding the gamma function and factorials work?
Is $x$ an integer? if not what does it mean $n \vert (x+1)$?
Apr
2
comment Best Fake Proofs? (A M.SE April Fools Day collection)
It seems that Euler himself was confused with this. See [here]( webspace.utexas.edu/aam829/1/m/Euler_files/EulerMonthly.pdf)
Jan
4
comment Exact power of $p$ that divides the discriminant of an algebraic number field
The statement will be easier to follow if you suffix the $\beta$'s. If I understand, there are $f_i$ elements in $B_i$, say $\beta_{ik}$ for $1\le k \le f_i$ and then the $n$ elements $\alpha_1,\dots,\alpha_n$ are $\alpha_{ij}\beta_{ik}$ for $1\le i\le r$, $1\le j\le e_i$ and $1\le k \le f_i$. In addition I think the $x \equiv \alpha_{ij} \pmod{Q_i^{e_i}}$ should read $x \equiv 1 \pmod{Q_i^{e_i}}$. If it is so, then the "large number $N$" means possibly: $N \ge e_i$ for every $i$ as you are then free to chose $\alpha_{ij}$ with the stronger constraint $\alpha_{ij} \equiv 0 \pmod{Q_h^N}$.
Dec
1
comment Galois group and the Quaternion group
I have corrected the right hand side of $(\theta^2-6-2\sqrt{3})^2$, sorry for that. I have also added some computations in the end to help you with the automorphisms.