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 $$\begin{aligned} \sum_{\substack{n\le a\\n\text{...
View ArticleComment by Pruthviraj on Is there any perfect square number can write in...
Thank you @DietrichBurde ,understand.
View ArticleComment by Pruthviraj on Uniqueness of negative and positive digit...
@lulu I Apologise for my English, it is distinct because Oder of sequence of negative postive place are distinct as, $23=(2,3)$ which have $(+,+)$ Oder and $23=(3,-7)$ have $(+,-)$ oder. I will try to...
View ArticleComment by Pruthviraj on Representing number, where digits are cubes
Thanks, how to use this code, when I plot it, gives, invalid literal for int() with base 10: '-f'
View ArticleComment by Pruthviraj on How to show relation between squarefree sets
@lulu $ij$ means $i$ times $j$. for example $A_2=\{2n:n\in\mathbb{N}\}$ and $A_3=\{3n:n\in\mathbb{N}\}$ then $A_2\cap A_3=A_6=\{6n:n\in\mathbb{N}\}$
View ArticleComment by Pruthviraj on How to show relation between squarefree sets
I'm sorry but i have change the direction of the last question can you please go with this because last problem is very unclear and have to specify it.
View ArticleComment by Pruthviraj on How to show relation between squarefree sets
thank you for your answer , Im on understanding this, I have just posted related Questions on mathoverflow [mathoverflow.net/q/462029/149083]
View ArticleComment by Pruthviraj on Properties of $\left[\frac{(mt+r)^m}{m}\right]$ in...
@DanielDonnelly Yes it is floor function. Motivation: maybe we can understand structure or pattern of digits for perticular expression. Actually I didn't tried much. I'm just excited with second...
View ArticleComment by Pruthviraj on Representing every positive integer using floor...
thank you so much, can we extend this to all integer power?
View ArticleShow that $-1^2+3^2-5^2\mp ...+(2^n-1)^2=2^{2n-1}$
Playing with numbers, I construct following expression.Can it be shown that$$\sum_{i=1}^{2^{n-1}}(-1)^i(2i-1)^2=2^{2n-1}$$attemptWe can construct following, using finite calculus...
View ArticleShow that the following inequality is true (about power sum)
Given $a$ and $b$ are non negative real number and $m\in\mathbb{R}_{\ge1}$Can it be shown that, If $b\ge a$ then$$\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...
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