there exist infinite many n∈Nn\in\mathbb{N} such that Sn−[Sn]<1n2S_n-[S_n]<\frac{1}{n^2}

Let Sn:=1+12+13+…+1n. Is it true that the set of n∈N such that Sn−[Sn]<1n2 is infinite? Here, [x] represents the largest integer not exceeding x. This question has been asked previously on math.SE without receiving any answers. Answer AttributionSource : Link , Question Author : math110 , Answer Author : Community

Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures

I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such that $\bmod(n, p) = A_p$ or $\bmod(n, p)= B_p$ with $0< A_p\leq B_p < p$. The choice of $M, A_p$ and $B_p$ is discussed later … Read more

Total coloring conjecture for Cayley graphs

The total coloring conjecture (TCC) states that any total coloring of a simple graph $G(V,E)$ has its total chromatic number bounded as $\chi^{T}(G)\le \Delta+2$ where $\Delta $ is the maximal degree of its vertices. The conjecture is open for many graphs albeit has been proved for several classes of graphs like complete graphs, trees, cycles, … Read more

What explains the asymptotic and the pattern in this sequence related to Riemann zeta zeros?

As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form: ℑ{ρn}=2πlogxn where xn is unknown. Therefore I solved for xn and got this integer sequence: a1(n)=(n+1)([1e2πℑ(ρn+1)−1]−[1e2πℑ(ρn)−1]) where [number] is the round or nearest integer function. This is a sequence starting: a1=2, 3, 0, … Read more

The action of the unitary divisors group on the set of divisors and odd perfect numbers

Let n be a natural number. Let Un={d∈N∣d∣n and gcd(d,n/d)=1} be the set of unitary divisors, Dn be the set of divisors and Sn={d∈N∣d2∣n} be the set of square divisors of n. The set Un is a group with a⊕b:=abgcd(a,b)2. It operates on Dn via: u⊕d:=udgcd(u,d)2 The orbits of this operation “seem” to be Un⊕d=d⋅Und2 for each d∈Sn From … Read more

Two questions on “Table problem on $\Bbb S^2$”

The following conjecture is known as “Table problem on $\Bbb S^2$” Conjecture (Table problem on $\Bbb S^2$): Suppose $x_1, x_2,x_3,x_4 \in\Bbb S^2 \subseteq \Bbb R^3$ are the vertices of a square that is inscribed in the standard $2$-sphere, and let $h : \Bbb S^2\to \Bbb R$ be a smooth function. Then there exists a rotation … Read more

A conjecture like Cayley–Bacharach theorem

Let six points A,A′,B,B′,C,C′ lie on a conic and a cubic. Let a conic through B,B′,C,C′ and meets the cubic again at A1,A2. Let a conic through C,C′,A,A′ and meets the cubic again at B1,B2. Let a conic through A,A′,B,B′ and meets the cubic again at C1,C2. Then six points A1,A2,B1,B2,C1,C2 lie on a conic. … Read more

The set of numbers a+ba+b such that ma2+nb2ma^2+nb^2 is prime

Conjecture: If m,n are coprime it exist a minimal natural number Nmn such that: {a+b>Nmn∣a,b∈N+∧ma2+nb2∈P>2}={k>Nmn∣gcd. Test for k\leq 10^8 suggest that N_{11}=1. The table below shows the largest minimal N_{mn} for k\leq 10^5. In the table 0 stands for \gcd(m,n)>1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 … Read more