Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a ...

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67 views

existence of elements with specific norms in pure cubic fields

Is there any specific way to find an element with a given norm in pure cubic field? say for an example an element of norm 5 in pure (monogenic) cubic field of 11. it is easy to check that 5 as an ...
4
votes
1answer
167 views

Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...
4
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2answers
314 views

Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture Let $f(x)$ be non-constant irreducible polynomial with integer coefficients, no fixed prime factor and positive leading coefficient. Let $S$ be a finite set of ...
0
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1answer
82 views

Joint Modular Distribution of Primes

Dirichlet's theorem shows that, for any fixed prime integer a, "big prime numbers mod a" are uniformly distributed between 1 and a-1. If we similarly pick different prime integers b,c,..., are these ...
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0answers
127 views

Updated tables of maximal prime gaps? [closed]

The website http://www.trnicely.net/gaps/gaplist.html contains a long list of maximal and nonmaximal prime gaps. In this list, the largest maximal prime gap is one of length 1476: Size: 1476 Gap ...
7
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1answer
325 views

Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms ...
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0answers
86 views

For which types of problems can one expect to use Bombieri-Vinogradov in place of GRH?

I must profess a general ignorance of problems that were once known to be true under GRH but has since became unconditional due to the Bombieri-Vinogradov theorem, but I am aware of the heuristic that ...
4
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1answer
156 views

Higher dimensional generalization of the Hardy-Littlewood conjecture?

The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h_1, \cdots, h_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a ...
15
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1answer
770 views

Sums of primes that are themselves prime

I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let: $$S(n) = \sum_{i=1}^n p_i$$ be the sum of the first $n$ primes and $$P(n) = | \{1 \leq i \leq n \mid ...
0
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1answer
142 views

Tail of singular series of Goldbach problem

Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ ...
13
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2answers
939 views

Can I express any odd number with a power of two minus a prime?

I have been running a computer program trying to see if I can represent any odd number in the form of $$2^a - b$$ With b as a prime number. I have seen an earlier proof about Cohen and Selfridge ...
4
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0answers
314 views

About sign changes of Li(x)-π(x)

Given a constant $C$, which are the best known upper bounds for the number of sign changes of the function $$ f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x) $$ in the range ...
17
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2answers
735 views

Floors of rationals to powers: Infinite number of primes?

Let $r=a/b$ be a rational number in lowest terms, larger than $1$, and not an integer (so $b > 1$). Q. Does the sequence $$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor, ...
11
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1answer
399 views

Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...
8
votes
1answer
426 views

Would Elliott-Halberstam conjecture follow from GRH?

The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...
2
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0answers
274 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} ...
14
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1answer
1k views

When is the product (1+1)(1+4)…(1+n^2) a perfect square?

This is a modification of an unanswered problem on the math StackExchange. When is the product (1+1)(1+4)…(1+n^2) a perfect square? If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n=3, k=10.$ ...
7
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2answers
325 views

convergence in Z-hat; modulo prime power

The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16). Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$ by $a_0=b, a_{n+1}=2^{a_n}$. Prove that ...
3
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2answers
194 views

Is there a lower bound for the first non-trivial sequence of consecutive integers where each of the first $n$ primes is a least prime factor

Using the Chinese Remainder Theorem, it is very straight forward to find a sequence of consecutive integers starting at $x$ where each of the first $n$ prime numbers is a least prime factor for a ...
8
votes
1answer
400 views

Is $\{ p \alpha \}$ for prime $p$ dense in $[0,1]$?

Let $\alpha$ an irrational real number. It is well known that the set $\{ \{n \alpha \}|\,\, n \in \mathbb{N} \}$ is dense in$[0,1]$. ($\{x\}$ denotes the fractional part of $x$) But how to prove the ...
4
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0answers
85 views

On a weighted sum in a lemma for sieve methods

I'm reading James Maynard's paper "Small gaps between primes". Lemma 6.1 (p.14) in this paper confused me. This lemma was taken from Goldston-Graham-Pintz-Yildirim's paper "Small gaps between ...
2
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1answer
170 views

Generalizations of Chen's theorem

The two famous theorems of Jingrun Chen, both with similar proofs, state (respectively) that all sufficiently large even numbers are the sum of a prime and an element of $P_2$, and that there are ...
3
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0answers
214 views

Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051) $$ f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p $$ for $\chi(n)$ the characteristic function of the primes: ...
7
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1answer
250 views

Mersenne almost primes

I asked earlier whether it can be proved that infinitely many elements of $P_n$ for some positive value of $n$ (here $P_n$ refers to the set of numbers with at most $n$ prime divisors). There I ...
2
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0answers
218 views

Brun's Theorem for twin primes and its generalization [closed]

Brun's Theorem given in 1919 ensures that the sum of the reciprocals of the twin primes converges. Do you know a different proof of this same result? Moreover, you know if the "generalization" of it ...
6
votes
1answer
270 views

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{ ...
4
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4answers
643 views

Prime numbers and limit ordinals

As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, ...
7
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1answer
501 views

Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq ...
1
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0answers
108 views

Which upper bound for $r_{0}(n)$ can be obtained through the Chinese Remainder theorem?

Assume Goldbach's conjecture. Then for every integer $n$ greater than one there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are prime numbers. For a given $n$, let's denote ...
19
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1answer
518 views

Avoiding multiples of $p$

Let $p$ be a prime number and $P=\{1,2,...,p-1\}$ In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$ only when we sum the last summand? For ...
-4
votes
1answer
143 views

$p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
21
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2answers
1k views

What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is $$ H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right) $$ The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...
5
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1answer
417 views

The number of distinct prime factors of $n\in\mathbb N$

Let $\omega(n)$ be the number of distinct prime factors of a natural number $n$. Note that $\omega(n)=0\iff n=1$, and that $\omega(24)=\omega(2^3\cdot 3^1)=2\ (\not = 4)$. (For more details, you ...
7
votes
2answers
337 views

Primes $p$ for which $pk+1$ is prime for small $k$ (or approximating Sophie Germain)

The twin prime conjecture says there are infinitely many pairs $p,p+2$ that are both prime, and although we still don't know whether it's true there's been a lot of progress recently showing that ...
11
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1answer
614 views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{p}} $$ Additional information: Since $$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{n}} ...
-6
votes
1answer
181 views

Is $2^{p}-1$ prime iff for $\frac{p-1}{2}$ odd positive integers $n$ below $p$, $(n+2)\vert (2^{p}+n)$? [closed]

As I was playing around with Mersenne numbers, and discovered the notion of Wagstaff prime going off Wikipedia, I started considering the sequence, for a given $odd$ prime number $p$, defined as ...
0
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0answers
194 views

Conjecture about distribution of primes in arithmetic progression

For my work, i need the following Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q-1<N.$ Then $\forall a\in\left[1,\, ...
7
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2answers
510 views

The equation $x^m-1=y^n+y^{n-1}+…+1$ in prime powers $x,y$

Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$? This question arose in the ...
5
votes
0answers
248 views

$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...
0
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0answers
217 views

On $n$-th prime $\pmod {n}$

Has it been proved or disproved that for any fixed $a\geq 1$ there are infinitelly many primes $p_n\equiv a\pmod{n}$? I believe i have proved that for every $a\geq1$ there are infinitelly many ...
5
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3answers
263 views

Quadratic residues and nonresidues of arbitrary patterns

Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$. There is an integer $a$ such that $\left( \frac{a}{p_1} ...
2
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0answers
210 views

Efficient ways to count primes satisfying Zhang's theorem

The theorem of Yitang Zhang states that there exist a finite $k \in \mathbb{N}$ such that there exist infinitely pairs of primes $(p,q)$ such that $|p - q| \leq k$. The statement that $k$ can be taken ...
6
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2answers
483 views

On the prime number theorem in arithmetic progression

The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$ In a similar manner ...
1
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0answers
122 views

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$. $\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...
1
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1answer
1k views

Calculating pisano periods for any integer

I recently stumbled across this SPOJ question: http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the pisano period of a number. After I researched my way through the web, I found ...
6
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0answers
231 views

Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...
2
votes
4answers
664 views

Product of exponents of prime factorization

Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example, $$p(5184) = p(2^6 3^4) = 24 \;,$$ $$p(65536) = p(2^{16}) = 16 \;.$$ Define $P(n)$ as the number of iterations ...
5
votes
1answer
127 views

Short lattice vectors orthogonal to a random vector

Let $N$ be some prime number. Suppose I draw $s$ elements $g_1,..., g_s$, where each $g_i\in [N]$ is taken uniformly from some interval $I_i$ of size, say $\sqrt{N}$. Is it possible to provide a ...
49
votes
4answers
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 ...
11
votes
1answer
516 views

Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?

While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction $$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$ Based on what ...