Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girardstates 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 $B$...
View ArticleHow to know if a random natural number is a probable semiprime?
Let that $n$ be a natural number 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...
View ArticlePrimes 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 paritiessuch that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.A few examples are:$2+4995825^k$ is prime for...
View ArticleFixpoints 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...
View ArticleReference 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...
View ArticleFermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are...
(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$$ b^{p-1} \equiv 1 \pmod{ p^m}...
View ArticleHow to check that a number probably/likely has a divisor having a specific...
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb...
View ArticleTuples of natural numbers with no mutual divisibility and large reciprocal sums
Standard apology in case this is something simple, as I'm not a number theorist.Let $E_1, \dots, E_n$ be disjoint finite sets of natural numbers, such that for any $a_1 \in E_1, \dots, a_n \in E_n$, we...
View ArticlePrimality testing by reversible computation using the prime number theorem
Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are...
View ArticlePrime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:$$\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), Selberg...
View ArticleObstacles to computing $\pi(n)$ in $O(n^{2/3-\epsilon})$ time
Edit: Apologies, as mentioned in the comments I failed to notice the analytic algorithms that take $O(n^{1/2+\epsilon})$ time, so this question doesn’t make much sense. It’s possible there is a...
View ArticleWhat keeps asymptotic Goldbach's conjecture out of reach of current technology?
Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and...
View ArticleIs there a two-variable prime-representing polynomial (in the sense of...
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...
View ArticleGeometric 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...
View ArticleClik here to view.
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...
View ArticleFully 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 <...
View ArticleIs there a statement in Presburger arithmetic about primes this simple...
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary...
View ArticleDiscovering patterns in data and methodologies used
I feel really dumb asking this but are there examples of any type of data where once a pattern/structure is discovered, the pattern is usually simple but the methodology used to discover that pattern...
View ArticleRelation between elements with fixed exponent over different...
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...
View ArticleThe smallest solution to $2^{2k}-1=\text{powerful}$
Integer is powerful if all the exponents in its factorization are at least $2$.Every powerful integer can be written in the form $a^2 b^3$.For odd $k$, define $F(k)=2^{2k}-1=(2^k-1)(2^k+1)$.This...
View ArticleCould I possibly exploit distinct odd primes raised to 6 to solve Exact Three...
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...
View ArticleCan 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,\dots...
View ArticleWhich 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 is...
View ArticleIs it possible to have square-free order(s) in $\mathbb{Z}^\times_N$?
Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random where...
View ArticleEffective 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...
View ArticleA possible variant of Zagier's one-sentence proof for Fermat's sum of two...
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...
View ArticlePrime Hadamard matrices
Assume that $n$ is a sufficiently large number. Is there a Hadamard matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$)...
View ArticlePower of primes
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ is the $n^\text{th}$ prime number and $\sigma(n)$ is the sum of the divisors of $n$. Consider the...
View ArticleProof 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...
View ArticleRoth's theorem for primes in a given arithmetic progression to a large modulus
Let $\mathbb{P}_{a, q}$ denote the set of primes congruent to $a$ modulo $q$. Are there any estimates for the number of $3$-Arithmetic Progressions in the set $\mathbb{P}_{a, q}\cap [1, X]$, where $(a,...
View Article
