All Questions
Tagged with nt.number-theory prime-numbers
1,808 questions
3
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1
answer
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Increasing sequences and Wieferich primes
We are trying to show that primes of the form $a(n)$ can't be
Wieferich primes.
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, ...
- 7,193
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 $A = n$, $B = 1$, $...
- 35.5k
5
votes
1
answer
727
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Sum of two squares and implication of Bunyakovsky conjecture
Bunyakovsky's conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...
7
votes
3
answers
1k
views
Values where infinite products of primes and composites are equal
Highly grateful for your help/steers on the following question (at the end):
Take the infinite product:
$$\displaystyle T(s) = \prod _{n=2}^{\infty } \left( \dfrac{{n}^{s}} {{n}^{s}-1}\right)$$
for ...
CommunityBot
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8
votes
1
answer
245
views
Asymptotic density of sums of consecutive primes
Call a positive integer respectable if it is a sum of consecutive prime numbers. For example, every prime numbers is respectable. So are $3+5=8$, $2+3+5=10$, $5+7=12$, $3+5+7=15$, $2+3+5+7=17$, $7+11=...
CommunityBot
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4
votes
0
answers
238
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Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
3
votes
2
answers
802
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Goldbach conjecture and the difference of two primes
The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes?
That is, if $2n$ is a sum of ...
3
votes
0
answers
175
views
Proof of when 3 is a cubic residue modulo primes
I have recently been learning about cubic characters, and the machinery of Gauss and Jacobi sums used to prove the cubic reciprocity theorem, and using this, I can now determine when any prime is a ...
- 2,811
6
votes
0
answers
381
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A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
1
vote
0
answers
128
views
Effective Erdős–Kac theorem
I have some number $N$ and some integer $k>0$. I want to know what fraction of numbers up to $N$ have more than $k$ prime factors. (In my application, with repetition, but the $\omega$ version is ...
- 9,114
-1
votes
3
answers
1k
views
Which even numbers are known to be both prime gaps and the sum of 2 primes?
Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
- 4,713
10
votes
2
answers
3k
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Can every integer be written as a sum of squares of primes?
This question is mainly inspired from a different problem I was working on.
Is there a value of $k$ such that, for each $n\in \mathbb N$, the equation
$$\sum_{i=1}^{k}x_i^2=n$$
is solvable in $x_1,\...
0
votes
1
answer
124
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Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?
I'm solving Exact 3 Cover, given a list with no duplicates $S$ of $3m$ whole numbers and a collection $C$ of subsets of $S$, each containing exactly three elements. The goal is to decide if there are $...
- 63.9k
0
votes
0
answers
98
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Relation between elements with fixed exponent over different $\mathbb{Z}^\times_p$
A primitive root $h$ of $n$ is a generator of the cyclic modulo multiplicative group $\mathbb{Z}^\times_n$.
Suppose, $\mathbb{P}_{\langle 2\rangle,N}=\{p_i <N\mid \langle 2\rangle=\mathbb{Z}^\...
3
votes
1
answer
329
views
Fully explicit Linnik's Theorem
Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
- 105k
19
votes
6
answers
37k
views
Pascal triangle and prime numbers
Back in the days when I was in high school, I developed a big interest about number theory specifically prime numbers and prefect numbers, I used to stay awake all night long with a bunch of sketch ...
5
votes
1
answer
750
views
Geometric mean of prime factors of all numbers up to n
Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...
- 105k
10
votes
2
answers
822
views
Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?
In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...
- 9,114
11
votes
2
answers
615
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
- 12.3k
2
votes
0
answers
121
views
How to know if a random natural number is a probable semiprime?
Let that $n\in\Bbb N$ generated from a hash function where $n$ is long enough to be hard to factor in the gnfs algorithm. How to check if $n$ is probably a semi‑prime in a faster way than factoring it ...
- 87
12
votes
2
answers
1k
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Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
- 105k
2
votes
0
answers
66
views
How to check that a number probably/likely has a divisor having a specific bit length/in range?
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
- 87
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votes
1
answer
652
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Fermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?
(I've taken this from MSE, it seems to be more appropriate here)
I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the
Question for
$$ b^{p-1} \equiv 1 \pmod{ ...
- 34.3k
25
votes
1
answer
911
views
Reference request for a proof of the two-square Theorem
One can show (see below for a sketch of a proof) that every odd prime number $p$
can be written in exactly $(p+1)/2$ different ways as
$$p=a\cdot b+c\cdot d$$
with $a,b,c,d\in\mathbb N$ satisfying $\...
- 105k
10
votes
1
answer
315
views
Fixpoints of $m\longmapsto \mathrm{rad}(\phi(m^2))$ under iteration
Given a strictly positive integer $m$ let $\alpha(m)=\mathrm{rad}(m\phi(m))$
be the radical (product of all distinct prime divisors) of the product of $m$ and of Euler's totient function $\phi(m)=m\...
- 105k
2
votes
0
answers
93
views
Primes of the form $a+b^k$ for $k=(a \bmod 2),\ldots,n$?
Procrastinal problem: Given $n$, one can ask for integers $a,b>1$ of different parities
such that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.
A few examples are:
$2+4995825^k$ is prime for $k=0,\...
- 105k
6
votes
1
answer
392
views
Arithmetic properties of positively reduced $2\times 2$-matrices
Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$
positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces
at ...
- 17.9k
1
vote
1
answer
214
views
Primes of the form $p_{i_1}p_{i_2}\cdots p_{i_n}+2k$
Let $S_{n,k}$ be the set of all numbers that can be written as the product of $n$ odd primes plus $2k$. Are there integers $n>1$ and $k>1$ such that $S_{n,k}$ contains finite number of primes?
- 4,713
5
votes
2
answers
691
views
Representing natural numbers as sums of distinct prime powers
I am investigating whether every natural number $n > 18$ can be represented as a sum $p_1^{m_1} + \dots + p_k^{m_k}$, where $p_1, \dots, p_k$ are distinct primes, and $m_1, \dots, m_k$ are distinct ...
- 105k
0
votes
1
answer
1k
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Alternative proofs of Euclid-Euler theorem
What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
4
votes
0
answers
200
views
Effective bound for odd numbers expressed as sums of three primes
I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...
CommunityBot
- 1
2
votes
0
answers
147
views
Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 105k
7
votes
1
answer
339
views
Rational prime factors in the components of powers of Gaussian primes
Let $\pi=a+bi\in \mathbb{Z}[i]$ be a Gaussian prime with $a$ and $b$ nonzero, and $b$ even. For odd rational primes $p=\pi\bar\pi$ and $q\neq p$, define $\pi^{\frac{1}{2}\left(q-\left(\frac{-1}{q}\...
- 148k
4
votes
0
answers
446
views
There are infinitely many prime which have arbitrary large gap in their digits in particular base expansion
Consider $m$ and $r$ is any fixed positive integer and $t$ is a variable $(t=0,1,2,3,...)$. Below, $[a]$ denotes the greatest integer function of $a$ (or floor function).
Claim 1 : There exists a ...
CommunityBot
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0
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Existence of minimal bases in additive combinatorics
Let $\mathbb{N}$ denote the set of natural numbers, including zero. A subset $X \subseteq N$ is a basis if $X + X = \mathbb{N}$. Clearly, if $X$ is a basis and $X \subseteq Y$, then $Y$ is also a ...
11
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1
answer
565
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Near-Legendre Conjecture
Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open....
- 294
2
votes
0
answers
124
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On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
- 12.6k
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 ...
- 3,319
8
votes
1
answer
863
views
On the least prime in arithmetic progressions
My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that
$$p(a, q) \ll q^L$$
for some ...
- 1,446
1
vote
1
answer
325
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Goldbach conjecture reformulation [closed]
As thought, the question below is a reformulation of the goldbach conjecture.
$ S = \{K - ap \mid a \geq 3, p \text{ is prime} < K/2 \} $, where $ a $ is an odd integer greater than or equal to 3, ...
- 23
0
votes
0
answers
122
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Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
6
votes
1
answer
478
views
How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?
Consider on the natural number the lexicographic ordering on the prime factorization:
If $m = p_1^{a_1}\cdots p_r^{a_r},n = q_1^{b_1}\cdots q_s^{b_s}$ then we define:
$$m \vartriangleleft n :\iff [(...
- 3,064
10
votes
1
answer
644
views
Independence between the number of prime factors of $n$ and $n+2$
I am interested in having an upper bound for the cardinality of
$\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$
for $k,\ell\geq 1$,
where $\omega(n)=\sum_{p\vert n}1$ counts the number ...
- 342
7
votes
0
answers
271
views
A question about prime numbers, totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $
This question was previously posted to MSE here.
I noticed something with the totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $ when $n > 1$.
It seems than :
$ \sigma(4n^2-1) \...
CommunityBot
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5
votes
2
answers
652
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On the number of distinct prime factors of $p^2+p+1$
Is it true that, for each positive integer $c$, there exists a prime number $p$ such that $p^2+p+1$ is divisible by at least $c$ distinct primes?
CommunityBot
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4
votes
0
answers
536
views
Is the integer factorization into prime numbers normally distributed?
Edit:
Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking.
Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, ...
- 3,064
5
votes
0
answers
131
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Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
- 51
8
votes
1
answer
664
views
The tightest prime zipper
Define a prime zipper as an increasing function $f(n)$ mapping $\mathbb{N}$ into $\mathbb{N}$
with the property that, for every $n \ge 1$, there is at least one prime within the
inclusive interval $[ ...
- 1,446
2
votes
2
answers
592
views
Is it true that there always exists a positive integer $n$ such that $p \mid \lfloor k^n\cdot\alpha\rfloor$?
Let $k,M$ be positive integers such that $k−1$ is not squarefree. Prove that there exist a positive real number $\alpha$, such that $\lfloor\alpha\cdot k^n\rfloor$ and M are coprime for any positive ...
CommunityBot
- 1
8
votes
1
answer
657
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Arithmetic sequences and Artin's conjecture
(Sorry if this is a naive question; it is not my area!)
Consider the following strengthening of Artin's conjecture on primitive roots (and Dirichlet's theorem) for the case of $n=2$: every arithmetic ...
- 79.9k