Comment by Pruthviraj on Show that,...
Thank you for your complete answer. Currently, I am a beginner in analytic number theory, can you please more explain the middle of answers portion $$\begin{aligned} \sum_{\substack{n\le a\\n\text{...
View ArticleComment by Pruthviraj on Is there any perfect square number can write in...
Thank you @DietrichBurde ,understand.
View ArticleComment by Pruthviraj on Uniqueness of negative and positive digit...
@lulu I Apologise for my English, it is distinct because Oder of sequence of negative postive place are distinct as, $23=(2,3)$ which have $(+,+)$ Oder and $23=(3,-7)$ have $(+,-)$ oder. I will try to...
View ArticleComment by Pruthviraj on Representing number, where digits are cubes
Thanks, how to use this code, when I plot it, gives, invalid literal for int() with base 10: '-f'
View ArticleComment by Pruthviraj on How to show relation between squarefree sets
@lulu $ij$ means $i$ times $j$. for example $A_2=\{2n:n\in\mathbb{N}\}$ and $A_3=\{3n:n\in\mathbb{N}\}$ then $A_2\cap A_3=A_6=\{6n:n\in\mathbb{N}\}$
View ArticleComment by Pruthviraj on How to show relation between squarefree sets
I'm sorry but i have change the direction of the last question can you please go with this because last problem is very unclear and have to specify it.
View ArticleComment by Pruthviraj on How to show relation between squarefree sets
thank you for your answer , Im on understanding this, I have just posted related Questions on mathoverflow [mathoverflow.net/q/462029/149083]
View ArticleComment by Pruthviraj on Properties of $\left[\frac{(mt+r)^m}{m}\right]$ in...
@DanielDonnelly Yes it is floor function. Motivation: maybe we can understand structure or pattern of digits for perticular expression. Actually I didn't tried much. I'm just excited with second...
View ArticleComment by Pruthviraj on Representing every positive integer using floor...
thank you so much, can we extend this to all integer power?
View ArticleShow that the following inequality is true (about power sum)
Given $a$ and $b$ are non negative real number and $m\in\mathbb{R}_{\ge1}$Can it be shown that, If $b\ge a$ then$$\left|\sum_{i=0}^{n-1}(-1)^i(a+ib)^m\right |\le\left|\sum_{i=0}^{n}(-1)^i(a+ib)^m\right...
View ArticleShow that there is no pair s.t. $a^1+a^2+a^3+...+a^n=b^m$
Let $a,b,m$ and $n$ belong to integer. Is there any pair exist such that$a^1+a^2+a^3+...+a^n=b^m$ where $a,b,m,n\ge 2$Source code...
View ArticleProblem on inequality with power sum
Let $m$ and $n$ are positive integerCan it be shown that, For every $m\ge 5$$$\sum_{i=1}^ni^m-m^i>0\iff n=2,3,...,m$$Example: let $m=5$, choose any $n$ between $2$ to $5$, now let $n=2$ then...
View ArticleFor all $t\in\mathbb{N}$, show that this sequence will eventually reach the...
Let $a_0=3,b_0=1$ and $t$ be an positive integer, define$$ a_{n+1} =\begin{cases}a_n+2\cdot t\cdot b_n-1, & \text{if $a_n$ is odd} \\a_n/2, & \text{if $a_n$ is even}\end{cases}$$$$...
View ArticleCan a sum of first consecutive $n$th numbers ever equal a power of three?
Define $S(n)=1+2+3+\dots+n$Note:$S(2)=3$Is the following claim true?For all $n>2$, there is no such $S(n)=3^t$ where $t$ positive integer?Small attempt, let $S(n)=\frac{n(n+1)}2=3^t$ then...
View ArticleInteresting thing about sum of squares of prime factors of $27$ and $16$.
Let$$n=p_1×p_2×p_3×\dots×p_r$$where $p_i$ are prime factors and$f$ is the functions$$f(n)=p_1^2+p_2^2+\dots+p_r^2$$If we put $n=27,16$ and $27=3×3×3$, $16=2×2×2×2$...
View ArticleProblem related to divisibility of even power sum
Define $S_m(n)=1^m+2^m+\cdots+n^m$Can it be shown that$S_{2m}(uv)\equiv0\pmod{uv}\iff S_{2m}(u)\equiv0\pmod{u}$ and $S_{2m}(v)\equiv0\pmod{v}$Where $m,u,v$ are positive...
View ArticleQuestion on next prime function
The function $P(n)$ gives the smallest prime larger or equal to n. Example: $P(3)=3,P(4)=5.$Show that: equation $P(x)^2-P(x^2)\equiv 4\pmod6$ have only one solution as $x=3$?Source code Pari...
View ArticleShow that there are only four solution to $P(x^2)\equiv -1\pmod{x}$
The function $P(n)$ gives the smallest prime larger or equal to n. Example: $P(3)=3,P(4)=5.$Show that, Equation $P(x^2)\equiv -1\pmod{x}$ have only four solution such as $x=1,2,3,5$? Where...
View ArticleAn Inequalities related to power sum
Let $S_m(n)=1^m+2^m+\cdots+n^m$Show that the following inequalities are true for all $m,k\in\mathbb{Z}_+$[1] $k\cdot(2km+m)^{2m-1}\le S_{2m-1}(2km+m-1)$[2] $k\cdot(2km+m+k+1)^{2m}\le...
View ArticleDoes parity of $f(a)$ and $f(a+1)$ are same whenever $a$ is even? Question...
Let $D(a, b)$ denotes sum of digits of $a$ in base $b$.Example: $D(5,2)=2,D(1227,10)=1+2+2+7=12$Define $f(a)=\sum_{i=2}^aD(a, i)$ where $a\ge2$.Example $f(5)=D(5,2)+D(5,3)+D(5,4)+D(5,5)=2+3+2+1=8$Can...
View ArticleShow that $\sum_{k=1}^{a-1}(k,a)\equiv0\pmod{a-1}\iff a $ is prime
Denote $\gcd(a,b)$ as $(a,b)$Let $a\ge 2$, can it be shown that$$(1,a)+(2,a)+\cdots+(a-1,a)\equiv0\pmod{a-1}$$Satisfy if and only if $a$ is prime?Clearly if $a$ is prime then...
View ArticleWhy sum of digits in even base have such property?
Function $D(a, b)$ define as sum of digit of $a$ in base $b$. Example $D(5,2)=2$.Let $$f(m, n)=\sum_{k=1}^m(-1)^{D(k, n)}$$Example...
View ArticleDivisibility criteria for 3 in sum of power of sum of digits
Following things motivated by previous post.Consider $a, b$ and $c$ are integer with $a\ge1$ and $b\ge 2$.Function $D(a, b)$ define as sum of digit of $a$ in base $b$. Example $D(5,2)=2$.Let...
View ArticleThere is no three consecutive integers with same Parity of sum of its digits
Let $D(a)$ denotes sum of digits of $a$ in decimal. Examples $D(49)=4+9=13$Let $P(a)$ denotes parity of $a$. Example $P(2)=0$ as even and $P(3)=1$ as odd.Questions: show that there is no $a$ such that...
View ArticleShow that, $(-1)^{\mu(1)}+(-1)^{\mu(2)}+...+(-1)^{\mu(n)}
Following is an experimental math claim.We denote $\mu(a)$ as Möbius functionLet $$F(a)=\sum_{i=1}^{a}(-1)^{\mu(i)}.$$Can it be shown that for every positive integer $a$,...
View ArticlePattern in alternating sum of cube be a square number
How to showLet$$S(n)=\sum_{k=1}^{2n^2-2n+1}(-1)^kk^3$$$\mid S(n)\mid $ is always be perfect square number for $n=1,2,3,...$.$\mid\sum_{k=1}^m(-1)^kk^3\mid$ is perfect square number if and only if...
View ArticleCan it be shown, $n^4+(n+d)^4+(n+2d)^4\ne z^4$?
We know, $n^4+(n+d)^4= z^4$ has no solution in positive integers $n,d,z$.Can it be shown, $n^4+(n+d)^4+(n+2d)^4= z^4$ has no solution in positive integers $n,d,z$?I am check upto $1\le n, d, z\le 150$...
View ArticleFormula for the first base-$n$ digit of $n^{d+1}-\sum_{i=1}^{n}i^d$
Let $k,n,d \in \mathbb{N}$ and $n>1$.Converting $k$ in a base $n$,$$k = (n_{ l} ~\cdots~ n_2~ n_1)_{n} \quad\text{where}\quad n_{l} \ne 0$$Show that, if$$k = n^{d+1} - \sum_{i=1}^n i^d$$ then$$n_l=n...
View ArticleAnswer by Pruthviraj for Representing sum of power in different way.
Theorem 1: Given $b_1,b_2,...,b_m,b_{m+1}$ finite integers. Denominator $d$ is smallest positive integer for $b_l$ integer coefficient. consider $n=dt+r$ then$$\sum_{0<l\le m+1}\binom{n}l a_l=...
View ArticleRepresenting sum of power in different way.
Definition . For $m ≥ 0$, the $m$th power-sum denominator is the smallest positiveinteger $d_m$ such that $d_m · (1^m + 2^m +···+ n^m) $is a polynomial in $n$ with integercoefficients.The first few...
View ArticleUnique Structure in base for powers $1,2$ and $3$
Let's $1<a\in\mathbb{N}$And $$A^{k}=\sum_{i=1}^{a}i^{k}$$Here $t $ is a number from any base $q$ can be converted in base $b$ written as$$(t)_{q}=(b_{r} b_{r-1} ... b_{2} b_{1})_{b}$$Now function...
View ArticleRepresenting number, where digits are cubes
Let $n$ is integer and we want to express it in terms of $10$ as $$n=R^3_k10^k+R^3_{k-1}10^{k-1} \cdots+ R^3_0$$ where $R_i\in \{\pm0,\pm 1,\pm 2,\ldots,\pm9\}$Example : $37=...
View ArticleQuestion similar to Collatz conjecture
Let $a_0$ be an positive integer, define$$ a_{n+1} =\begin{cases}a_n/2, & \text{if $a_n$ is even} \\a_n+2a_0-1, & \text{if $a_n$ is odd}\end{cases}$$Now form a sequence $S(a_0)_{n\ge 0}$ by...
View ArticleRepresenting every positive integer using floor function and square number.
For a real number $x$, $\lfloor x\rfloor=\max\{n\in\mathbb{Z}\mid n\leq x\}.$Question : Every positive integer can be expressed as $\left\lfloor\frac{(mt+r)^2}{m}\right\rfloor$ for some positive...
View ArticleObservation on Erdős–Moser equation.
Define: $ S(n,m)= \sum_{i=1}^{n}i^m$ where $n,m\in \mathbb{Z}_+$Define: $F_m$ is function as, there exist smallest integer $k$ with respect to $m$ such that, $$k^m\le S(k-1,m)$$ so $F_m=k$.Example:...
View Article
