All Questions
Tagged with elementary-number-theory summation
482
questions
4
votes
1
answer
368
views
How many numbers less than N have a prime sum of digits?
I'm working on solving Project Euler's Problem 845. It's asking us to find the $10^{16}$-th positive integer number that has a prime sum of digits. Adopting a 'naive' solution, I compute the sum of ...
2
votes
2
answers
141
views
How to choose which stronger claim to prove when proving $\sum_{i=1}^n \frac{1}{i^2} \le 2$?
I am studying an inductive proof of the inequality $\sum_{i=1}^n \frac{1}{i^2} \le 2$. In the proof, it was decided to prove the stronger claim $\sum_{i=1}^n \frac{1}{i^2} \le 2-\frac{1}{n}$, as this ...
- 3,019
4
votes
1
answer
106
views
Simplifying floor function summation $\sum^m_{k=0} \left(\left\lfloor{\frac{n}{m}k}\right\rfloor-\left\lceil{\frac{n}{m}(k-1)}\right\rceil\right)$
Is there a way this summation can be simplified/cast in a more revelatory form (non-summation representation, single-term only, etc.)?
$$\sum^m_{k=0} \left(\left\lfloor{\frac{n}{m}k}\right\rfloor-\...
- 315
26
votes
1
answer
908
views
Mysterious sum equal to $\frac{7(p^2-1)}{24}$ where $p \equiv 1 \pmod{4}$
Consider a prime number $p \equiv 1 \pmod{4}$ and $n_p$ denotes the remainder of $n$ upon division by $p$. Let $A_p=\{ a \in [[0,p]] \mid {(a+1)^2}_p<{a^2}_p\}$.
I Conjecture
$$\sum_{a \in A_p } a=\...
- 3,792
2
votes
2
answers
354
views
At most one representation as a sum of two fibonacci-numbers?
I wanted to start a project to find primes of the form $F_m+F_n$ with integers $m,n$ satisfying $1<m<n$ , where $F_n$ denotes the $n$ th fibonacci-number.
I wondered whether duplicate numbers ...
- 85.1k
5
votes
2
answers
284
views
The most elementary proof of divisibility of sum of powers
I am wondering how one can prove that for an arbitrary odd natural number $n$ and an arbitrary natural number $a$ the power-sum $S_{(a,n)}=1^n+2^n+\ldots +a^n $ is divisible by $S_{(a,1)}=1+2+\ldots + ...
- 1,066
0
votes
2
answers
85
views
Construct a prime using $[2, 2, 2, ...., 3]$
Is there a generalized method to constructing primes through sums using the set $[2, 2, 2, ..., 3]$ given its elements are $n$- many 2s and a 3. This question obviously requires knowledge on ...
user1107557
1
vote
0
answers
65
views
Prove that $\lim\limits_{n\to\infty} a_n=\infty$.
Let $a_n$ denote the exponent of $2$ in the numerator of $\sum_{i=1}^n \dfrac{2^i}{i}$ when written in the lowest form. For example, $a_1=1,a_2=2,a_3=2.$ Prove that $\lim\limits_{n\to\infty} a_n=\...
- 1,837
2
votes
0
answers
71
views
When is $\sum_{1 \leq n \leq k}n^{-n+k}$ prime?
Consider the following finite sum $f: \mathbb{N} \rightarrow \mathbb{N}$ defined as
$$f(k) = \sum_{1 \leq n \leq k}n^{-n+k}$$
$$ = 2 + 2^{k-2} + 3^{k-3}...+ (k-1)$$
It is easy to see that $f(2) = 2$ ...
user1107557
2
votes
1
answer
127
views
Show that the product of the $2^{2019}$ numbers of the form $\pm 1\pm \sqrt{2}\pm\cdots \pm \sqrt{2019}$ is the square of an integer.
Show that the product of the $2^{2019}$ numbers of the form $\pm 1\pm \sqrt{2}\pm\cdots \pm \sqrt{2019}$ is the square of an integer.
I'm aware very similar problems were asked before (e.g. here and ...
- 2,031
1
vote
1
answer
314
views
Number theory approach to Project Euler's "Large Sum" problem?
I am refreshing some of my skills by solving problems on the Project Euler site. It is a repository of problems that usually require some mathematics knowledge and programming knowledge to solve ...
- 1,876
1
vote
0
answers
85
views
Probability that two integers are coprime
Maybe it is a silly question, but anyway. We know that the probability that two positive integers are coprime is $6/\pi^2$. However, for fixed positive integers $r$ and $s$, I'd want to compute the ...
- 73
6
votes
0
answers
57
views
For which integers $a\gt 0, b\ge 0$ is $\sum_{k=1}^n \frac1{ak+b}$ never an integer for $n > 1$?
For which integers $a\gt 0, b\ge 0$ is
$\sum_{k=1}^n \frac1{ak+b}$
never an integer for $n > 1$?
This is inspired
by a number of cases where this is true
($(a, b)
=(1, 0),
(3, 1)
$).
It might be ...
- 108k
6
votes
1
answer
228
views
An alternating sum
I ran into an alternating sum in my research and would like to know if the following identity is true:
$$
\sum_{i = 0}^{\left\lfloor \left(n + 1\right)/2\right\rfloor} \frac{\left(n + 1 - 2i\right)^{n ...
- 147
1
vote
1
answer
129
views
Proving $\sum_{1}^{n} \left\lceil\log_{2}\frac{2n}{2i-1}\right\rceil=2n -1 $
Show that $$\sum_{i=1}^{n} \left\lceil\log_{2}\frac{2n}{2i-1}\right\rceil=2n -1 $$ where $ \lceil\cdot\rceil$ denotes the ceiling function.
My method: one way would be observe each part of the ...
- 977