All Questions
36 questions
1
vote
0
answers
100
views
Curious congruences modulo $4$ involving primes
We define
$$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2}
\sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$
(Searching the OEIS yielded no results.)
For $n>2$ we have the following experimental observations (...
- 17.9k
4
votes
1
answer
240
views
Do there exist prime numbers of the form $n \cdot 2^n + 1$, when $n \in \mathbb{N}$ and $n > 1$?
Recently, I was studying prime sequences of the form $k \cdot 2^n + 1$, and I noticed that primes of the form $n \cdot 2^n + 1$ almost do not exist, except for the $n = 1$ case.
Are there other prime ...
- 125
2
votes
0
answers
76
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
1
vote
0
answers
60
views
On the parity of $(2^{\varphi(n)}-1) \bmod{n^2}$
For odd integer $n$ define the function
$$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$
$J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$.
Integer $n$ is Wieferich number
iff $J(n)=0$ and if $n$ is ...
- 25.4k
5
votes
1
answer
172
views
On vanishing of $p$-adic logarithms
Might be related to Wieferich primes.
Let $p$ be odd prime and define the Fermat quotient
$$F(n)=\frac{(2^{n-1} -1)}{n} \mod n=\frac{(2^{n-1} \bmod n^2 )-1}{n}$$
For integer $b$ let $L_p(b)$ be the $p$...
- 25.4k
2
votes
0
answers
156
views
Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
- 25.4k
6
votes
1
answer
393
views
Test for pair of odd primes $(p, 2p^2-1)$
Let $a(n)$ be A106483 (i.e., primes $p$ such that $2p^2-1$ is also prime).
Let $b(n)$ be an integer sequence such that $b(n) = B$ after the whole transformation where we start with $A = n$, $B = 1$, $...
- 4,938
2
votes
2
answers
197
views
On the primality of $j(n)=\varphi(p_n+1-n)+1$ when $j(n) \equiv 19 \pmod {100}$
Related to Power of primes.
Let $p_n$ denote n-th prime and $\varphi$ the totient function.
For natural $n$, define $j(n)=\varphi(p_n+1-n)+1$.
For $n$ up to $10^9$ if $j(n) \equiv 19 \pmod {100}$
then ...
- 25.4k
13
votes
1
answer
700
views
When is $\mathrm{gcd}(k,p^k-1)=1$ true?
Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$?
For the ...
- 63.1k
2
votes
0
answers
199
views
Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 4,938
4
votes
0
answers
306
views
How to explain this number-theoretic seeming “almost coincidence”?
For natural numbers $n\geq2$, let $d(n)$ be the number of divisors of $n$, and let
\begin{equation}
g(n)=n\sum_i r_i(p_i-1)
\end{equation}
where $n=\prod_i p_i^{r_i}$ is the factorisation of $n$ as a ...
- 141
3
votes
1
answer
2k
views
What do we know about Lucky numbers?
I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics.
Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
- 39
0
votes
1
answer
104
views
Non-Wieferich primes with Euler quotient modulo $p$ two and alternating harmonic numbers
Let $b(n)$ denote the Euler quotient modulo $n$.
In OEIS we have A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2)
For $n>1$ we have $b(A128465(n))=2$.
...
- 25.4k
0
votes
0
answers
110
views
What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?
Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...
- 25.4k
1
vote
1
answer
181
views
On the sequence $a(n)=\gcd(2^n-1,\phi(2^n-1))$
For natural $n$, define the sequence
$$
a(n)=\gcd(2^n-1,\phi(2^n-1))
$$
It doesn't appear to be in OEIS and starts
$1,1,1,1,9,1,1,1,3,1,9,1,3,1,1,1,27,1,75,49$
Q1 Can we unconditionally prove $a(n)=1$...
- 25.4k
1
vote
0
answers
153
views
A definition related to pseudoprimes and the Dedekind psi function
In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
4
votes
1
answer
219
views
Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$
Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
- 5,470
2
votes
0
answers
108
views
How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?
Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...
- 523
0
votes
1
answer
492
views
New experiments involving Ramanujan primes: Benford's law
I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the Ramanujan Prime from the online encyclopedia Wolfram MathWorld. This week I ...
2
votes
1
answer
220
views
Euler quotients modulo $n$
For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
$a(n)=0$ for OEIS sequence Wieferich numbers
...
- 25.4k
4
votes
1
answer
434
views
Quadratic progressions with very high prime density
In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this:
The list of sequences $f_k(x)$, each one corresponding to a specific
$k$, ...
- 3,098
2
votes
0
answers
326
views
Why can one compute the sum of divisors of $n$ without factoring $n$?
Question links to paper
which states:
$$
\sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1)
$$
where $\sigma(n)$ is the sum of divisors of $n$.
Another similar ...
- 25.4k
24
votes
1
answer
2k
views
Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 429
1
vote
0
answers
151
views
On smoothness and roughness of a number related to triangular numbers
Define $\triangle_n$ to be the $n$th triangular number.
Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$
Define $(\ell,k)$-smough numbers to be numbers that ...
- 1,826
4
votes
0
answers
178
views
Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers
The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by
$$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$
and
$$L_0=2,\ L_1=1,\ \text{...
- 15.6k
5
votes
0
answers
161
views
Consecutive integers each of which has a large prime factor
There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor?
More precisely, let $P(n)$ be the ...
- 341
1
vote
0
answers
223
views
Does each prime $p>3$ have a quadratic nonresidue which is a Mersenne number?
Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime.
QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?...
- 15.6k
5
votes
0
answers
176
views
Can the integers in an easily computable sequence free of prime numbers always be factored easily?
Call a sequence $(a_n)$ of positive integers easily computable
if there is a constant $C$ and an algorithm which computes $a_n$ from
$n$, $a_1, \dots, a_{n-1}$ and a finite number of integer ...
- 19.6k
26
votes
1
answer
3k
views
A surprising conjecture about twin primes
Just for fun, I began to play with numbers of two distinct ciphers. I noticed that most of the cases if you consider the numbers $AB$ and $BA$ (written in base $10$), these have few common divisors: ...
- 363
69
votes
1
answer
4k
views
Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?
I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime ...
- 34.3k
2
votes
1
answer
154
views
GCD for two Cullen numbers
The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
- 21
8
votes
1
answer
567
views
Arbitrarily many primes in a Fibonacci-type sequence
It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version:
Question. For any $K$, ...
- 765
15
votes
0
answers
487
views
Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
- 17k
2
votes
0
answers
311
views
A question concerning the strange arithmetic derivation
This question is related to Strange (or stupid) arithmetic derivation. The original question whether an unbounded sequence of iterates exists is still unanswered.
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \...
- 432
49
votes
4
answers
4k
views
Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
- 3,004
3
votes
3
answers
696
views
For any prime $p$, is there $C$ such that if $x\ge C$, then all but one integer among $x+1, x+2, \dots, x+p$ has Greatest Prime Factor $> p$
I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $\mathrm{gpf}(x) \le p$ where $p$ is any prime.
Clearly, as $x$ ...
- 1,049