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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
27
comment Need suggestions for this real world problem
The kind of problem that you are trying to solve is the Vehicle Routing problem. It's an NP Hard problem which is usually solved using heuristic searches and techniques like constrained integer programming.
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.
Nov
10
comment Early history of lower bounds on the prime counting function
I'm sorry I've made an incorrect statement, I could not edit it so I have removed it.
Nov
6
comment What would be complexity of computing $3^{n^n}$?
If you are computing $3^{n^n}$ modulo an integer $M$ and you know it's totient function $\varphi(M)$ then you can compute $n^n$ in $O(\log n)$ multiplications $\pmod{ \varphi(M)}$ and then rise $3$ to the result in $O(\log M)$ multiplications mod $M$ finding the result in polynomial time. The problem is to find $\varphi(M)$. Some times it can be done in polynomial time (for example if $M$ is prime then $\phi(M)=M-1$), but in general if $M$ is composite finding $\varphi(M)$ is equivalent to finding the factorization of $M$.
Nov
5
comment Proving a simple inequality
I find $$f(3) = \frac{\tfrac{2}{3}\log 2}{\log 6} = 0.2579... > f(4) = \frac{\tfrac{1}{2}\log 3}{\log 12} = 0.2210...$$ Are you sure the inequality is correctly written?
Nov
1
comment Half the rationals?
A good candidate for your set $X$ with a very simple description is the set of reduced fractions $a/b$ with $$ a\cdot b \equiv 0 \pmod{3} $$
Nov
1
comment Half the rationals?
I think there something missing in the right hand side of the inequality $\sum_{k=1}^{n-1} a_k\phi(k) < 1/2$, I suppose you mean $<\frac{1}{2}\sum_{k=1}^{n-1} \phi(k)$? On the other hand, even if it is probably true, I can't see how to prove that this construction works for all the intervals $Y$.