All Questions
42
questions
3
votes
0
answers
59
views
Is it possible to construct a sequence using the first $n$ prime numbers such that each segment has a unique sum?
For example, consider the sequence $2,7,3,5$. The sums of the segments of this sequence are as follows, and they are all unique:
$$2, 2+7, 2+7+3, 2+7+3+5, 7, 7+3, 7+3+5, 3, 3+5, 5$$
Can we generate ...
- 1,451
3
votes
0
answers
75
views
Can we show that $\frac{\sum_{j=1}^n j^2\cdot j!}{99}$ generates only finite many primes?
Define $$f(n):=\frac{\sum_{j=1}^n j^2\cdot j!}{99}$$
Is $f(n)$ prime for only finite many positive integers $n\ge 10$ ?
Approach : If we find a prime number $q>11$ with $q\mid f(q-1)$ , then for ...
- 85.1k
0
votes
1
answer
39
views
Differences between sums of reciprocals of primes and products thereof.
I am not a number theorist, but I know that number theorists are notorious for estimating sums including primes. So, I have the following question, which I may not have seen addressed, but I am sure ...
- 717
1
vote
0
answers
120
views
is a closed form solution possible?
I have a facination with prime numbers and unsolved problems, and i ended up creating a formula that returns the index of any prime number. The main problem is I don't know if the summations if its ...
- 23
2
votes
1
answer
66
views
Evaluate the sum or asymptotic expansion of $\sum_{p\ge 3} \frac{\log \left({p}\right)}{{p}^{2} - 1}$
In my research I cam across this sum
$$S = \sum_{p\ge 3} \frac{\log \left({p}\right)}{{p}^{2} - 1}$$
where $p$ is a prime number. Numerical evaluation of the first $10^8$ primes yields $S = 0....
- 967
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
$$\sum_{a \in A_p } a=\...
- 3,792
0
votes
2
answers
85
views
Construct a prime using $[2, 2, 2, ...., 3]$
Is there a generalized method to constructing primes through sums using the set $[2, 2, 2, ..., 3]$ given its elements are $n$- many 2s and a 3. This question obviously requires knowledge on ...
user1107557
1
vote
0
answers
85
views
Probability that two integers are coprime
Maybe it is a silly question, but anyway. We know that the probability that two positive integers are coprime is $6/\pi^2$. However, for fixed positive integers $r$ and $s$, I'd want to compute the ...
- 73
1
vote
0
answers
69
views
Minimum value made of the reciprocals of the first $n$ primes
Let $n$ be a positive integer , $p_k$ the k-th prime number and $a_j=-1$ or $a_j=1$ for $j=1,\cdots ,n$
What is the minimum value of $$S:=|\sum_{j=1}^n \frac{a_j}{p_j}|$$ ?
Motivation : If we ...
- 85.1k
5
votes
0
answers
86
views
How many solutions for $\,\sum_{k=1}^{n}p_k=p_m\cdot p_{m+1}\,$?
I ask for which pairs $(m,n)$ is satisfied
$$\sum_{k=1}^{n}p_k=p_m\cdot p_{m+1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$$
where $p_k$ is the $k$-th prime.
Up to $n=10^7$ I have found only the solution $(4,...
- 1,828
0
votes
1
answer
39
views
Indicator equation $f(n) = n$ for $n \in \Bbb{N}$ such that $n \pm 1$ is a pair of twin primes.
Consider the formula:
$$
f : \Bbb{N} \to \Bbb{C} \\
f(n) := 2 + \sum_{a = 1}^{n-2} \exp\left({\dfrac{2\pi i}{\gcd(a, n^2-1)}}\right)
$$
If $n \pm 1$ is a pair of twin primes, then $f(n) = n$. This is ...
2
votes
1
answer
449
views
Sum of all divisors of the first $n$ positive integers.
Yesterday I was going through Möbius Function notes, and found that
writing $n = p_{1}^{\alpha_1}p_{2}^{\alpha_2}\cdots p_{r}^{\alpha_r}$,
the sum of all divisors can be written as.
$$
e(n) = \prod_{...
- 49
2
votes
1
answer
63
views
For any prime $p>3$ show that $C_{np}^{p}-C_{np}^{2p}+C_{np}^{3p}-C_{np}^{4p}+...+(-1)^{n-1}C_{np}^{np} \equiv 1\pmod{p^3}$
Let $n$ be a positive integer. For any prime $p>3$ show that
$$C_{np}^{p}-C_{np}^{2p}+C_{np}^{3p}-C_{np}^{4p}+...+(-1)^{n-1}C_{np}^{np} \equiv 1\pmod {p^3}$$ Where $C_{n}^{k}=\frac{n!}{k!(n-k)!}$. (...
- 11.6k
4
votes
1
answer
194
views
Egyptian fractions with prime power denominators summing to 1?
Inspired by
On the decomposition of $1$ as the sum of Egyptian fractions with odd denominators - Part II
Can we solve the equation $$1=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots +\frac{1}{...
- 85.1k
7
votes
0
answers
183
views
Sum of numbers $x+y$ satisfying $x^2+y^2=p$ with $p$ being a prime number
(This question has been inspired by a similar one from James Johnson)
It's a well known fact that every prime number $p$ such that:
$$p\equiv 1 \pmod 4\tag{1}$$
can be represented as a sum of two ...
- 16k