All Questions
Tagged with prime-factorization prime-numbers
194
questions with no upvoted or accepted answers
31
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0
answers
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Have I discovered an analytic function allowing quick factorization?
So I have this apparently smooth, parametrized function:
The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$.
It is then defined for real $x$ apart from ...
- 1,091
13
votes
0
answers
886
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Whether there is a prime in sequence $\{1,12,123,1234,12345,123456,1234567,12345678,123456789,12345678910, \cdots\}$
UPD: To make it clearer, here is a statement:
For sequence $A = \{i\in\mathbb{N}_+\mid a_i=a_{i-1}\times 10^{\lfloor(\lg(10n))\rfloor} + n\}$, where $\lg n=\log_{10} n$, show whether there is a prime ...
- 305
10
votes
1
answer
289
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$pq\equiv 1\pmod 4$, how to find $p,q\bmod 4$?
Somebody asked me a question, I have no idea about it, the question is:
If a positive integer $n\equiv 1\pmod 4$ is the product of two primes, (denotes $n=pq,$ such as a RSA number) but we don't know ...
- 15.9k
9
votes
1
answer
195
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Prime Concatenation Order
Consider the following procedure.
Given an integer $n \geq 2$, obtain the canonical prime factorization of $n$, i.e. $\prod_{i=1}^k p_i^{e_i}$. Take the distinct factors $p_i$ and list them in ...
- 219
8
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182
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Odd numbers with $\varphi(n)/n < 1/2$
The topic was also discussed in this MathOverflow question.
From $\varphi(n)/n = \prod_{p|n}(1-1/p)$ (Euler's product formula) one concludes that even numbers $n$ must have $\varphi(n)/n \leq 1/2$ ...
- 18.2k
8
votes
0
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447
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The Greatest Common Divisor of All Numbers of the Form $n^a-n^b$
For fixed nonnegative integers $a$ and $b$ such that $a>b$, let $$g(a,b):=\underset{n\in\mathbb{Z}}{\gcd}\,\left(n^a-n^b\right)\,.$$ Here, $0^0$ is defined to be $1$. (Technically, we can also ...
- 49.8k
8
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0
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143
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Which prime factors of $8^{8^8}+1$ are known?
We have the partial factorization
$$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$
The first factor is $F_{24}$. It is composite, but no prime factor is known.
A prime factor of the second ...
- 85.1k
8
votes
0
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277
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How to list the prime factorised natural numbers?
Today I set out to invent a two character numeral system designed to make factorization trivial. Indeed, it lets one factor non-trivial numbers with over thousand digits within 30 seconds per hand - ...
- 12.3k
7
votes
0
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174
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Fibonaccis and prime numbers
Let $F_n$ denote the $n$th Fibonacci number, and $p_n$ the $n$th prime.
Let $a(n)$ be the smallest positive integer such that $p_n$ is a factor of $F_{a(n)}$.
How can I see that it follows that ...
7
votes
0
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93
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Prove there are infinitely many integer solutions to $z^z = y^y x^x$ for with $x,y,z > 1$
I have tried a number of methods using prime factorisations but they inevitably lead to invoking too many unknowns for me and balloon in complexity.
- 624
6
votes
0
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181
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Curiosity: $\text{antiprime} = \text{prime} + 1$
The following is just a mathematical curiosity that popped into my head that I thought was interesting. I haven't been able to find anything about it online, although maybe I just am unaware of the ...
- 555
6
votes
2
answers
123
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Proving that there is an infinite number of pairs of prime numbers for which $F(n)F(n+1) =pq $ for no $n>1 \in \mathbb{N}$, $F(n)$ is the GPF function
Proving that there is an infinite number of pairs of prime numbers for which $F(n)F(n+1) = pq $ does not hold for any $n>1 \in \mathbb{N}$, $F(n)$ is the GPF function
I have been trying to solve ...
- 151
6
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0
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200
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A conjecture regarding the sum of a number and its prime factorization
I am not sure if this a known problem, but I have a conjecture regarding the prime numbers. Since I don't know much analytic number theory, I thought maybe someone here could prove/disprove it.
Let $n\...
- 2,694
6
votes
0
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383
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What is the most frequent largest prime factor of the numbers between two primes?
Let $p_n$ be the $n$-th prime and $l_n, n \ge 2$ be the largest of all the prime factors of the composite numbers between $p_n$ and $p_{n+1}$. Since there are infinitely many prime gaps, each of these ...
- 20.8k
6
votes
0
answers
175
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Prime number theorem for knots?
Is there an analog to the prime-number theorem
describing the distribution of the prime numbers among the integers:
A theorem that describes the distribution of the prime knots,
perhaps with respect ...
- 30.4k