3
votes
0answers
24 views

Probability and sums of prime factors

Of the first N natural numbers, we select two different numbers at random. We'll call the greater one A and the lesser one B. What is the probability (P) that the sum of A's prime factors is LESS than ...
3
votes
1answer
70 views

Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
3
votes
0answers
89 views

Sum of product partitions of divisors

Let $M(n)$ be the the set of the multiplicative partitions of $n$, and let $D(n)$ be the set of the sum of the multiplicative partitions of the divisors of $n$. eg $M(30)=\{\{30\},\{2,15\},\{3, ...
1
vote
2answers
83 views

Dice-Game with two-twenty sided dice.

EDIT: I'll give this another try, trying to be clearer. The game is played like this: Player A roles two-twenty-sided dice and multiplies the two integers together to get some integer, say x, with $ ...
7
votes
0answers
202 views

Cramér's Model - “The Prime Numbers and Their Distribution”

I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start ...
2
votes
0answers
38 views

Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let $$ h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}| $$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = ...
1
vote
0answers
59 views

Using Bayes' Theorem to calculate if a number is prime if it passed a primality test

According to this, I can use Bayes theorem to calculate if $n$ is composite given that it passed the Miller-Rabin primality test: ...
5
votes
1answer
100 views

relative size of most factors of semiprimes, close?

when chatting about RSA a cohort just asserted something like "most prime factors of semiprimes are roughly the same size" measured in bits. ie "bits" is the number of digits in the base2 ...
2
votes
1answer
27 views

Deriving the equation for the probability of a prime number

A couple of months ago I stumbled upon an equation regarding the probability of having a prime number adjacent to "x". If I remember correctly it was: $\frac {x}{ln x}$ ,or something along those ...
2
votes
1answer
100 views

Probability property that the longest side of primitive Pythagorean triples is prime

If we consider the set of the first $n$ primitive Pythagorean triples, then the probability that the triple's longest side is prime is approximately $\dfrac{1}{\log_{11.475}n}$ based on Mathematica’s ...
0
votes
2answers
101 views

Primes dividing Integers

I randomly choose two integers. What is the probability that a certain prime number p does not divide both integers? Express your answer in terms of p.
11
votes
2answers
589 views

Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
2
votes
0answers
54 views

Merthen's third theorem and uncertainty of prime hits

Conjecture(1) Merten's third theorem says: $$\lim_{L\to\infty}\ln L\prod_{p\le L}\left(1-\frac1p\right)=e^{-\gamma}$$ we have a wild discussion here around the table whether it is possible to ...
3
votes
0answers
84 views

A challenging problem on prime uncertainty interval

I have a very challenging problem to solve, seeking for good advice; I have to make an intro in the first part and then comming to the problem. Theorem (1): In an interval between a prime $p$ and its ...
1
vote
1answer
90 views

What is the probability of $p_n$ being the greatest prime factor of a random number?

What is the probability of $p_n$ being the greatest prime factor of a random number?
1
vote
3answers
75 views

Questions about assigning a probability to a randomly chosen large integer $n$ being prime

I heard this question a few days ago, so reciting from memory: If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
5
votes
1answer
162 views

P[random x is composite | $2^{x-1}$ mod $x = 1$ ]?

Select a uniformly random integer $n$ between $2^{1024}$ and $2^{1025}$ (Q) What is the probability that n is composite given that $2^{n-1}$ mod $n = 1$ ? How did you calculate this? More info: ...
7
votes
3answers
196 views

Generating random numbers with the distribution of the primes

I would like to generate random numbers whose distribution mimics that of the primes. So the number of generated random numbers less than $n$ should grow like $n / \log n$, most intervals ...
3
votes
2answers
126 views

Why is the probability that a prime p is a factor of a number n equal to 1/p

I'm learning some number theory and I can't seem to understand why this is the case.
3
votes
3answers
349 views

Expected smallest prime factor

For a random integer $x$ chosen uniformly between 2 and $n$, what is the expected value of the smallest prime factor of $x$ as a function of $n$? What is the behavior of the function as $n$ tends to ...
0
votes
0answers
100 views

Is this probabilistic argument about “mutual primitive roots” correctly done?

A recent sci.math thread is called "mutual primitive roots". It is about quasi's conjecture that For each prime $q>2$, there is a prime $p<q$ such that $p$ is a primitive root of $q$, and ...
4
votes
1answer
89 views

What is the probability of picking a random prime < n?

If I shoot in the dark and pick a random number that's $<n$, what's the probability that the number will be prime? How many guesses, on average, would it take to get a prime number? I would really ...
2
votes
1answer
103 views

With what probability is this polynomial equal to zero (mod a prime $p$)?

If we suppose that we have a polynomial $q(x)$ of the following form: $q(x) = \sum_{i=0}^N{c_i x^i} \text{ where } c_i=0 \text{ or } c_i=1$ In other words, if we are given a polynomial with binary ...
7
votes
2answers
231 views

Given $2$ randomly chosen integers $x,y$ what is $P(k=gcd(x,y))$?

Given $2$ randomly chosen integers $x,y$ what is the probability that a integer $k$ is the greatest common divisor of $x$ and $y$? I know that the probability of $x,y$ being coprime is ...
12
votes
2answers
654 views

Ulam spiral: Is there an “unusual amount of clumping” in prime-rich quadratic polynomials?

I was reading Martin Gardner's Mathematical Games column on the Ulam spiral which appeared in the March 1964 issue of Scientific American. (The spiral actually featured on the cover of that issue.) ...
11
votes
1answer
678 views

What's the probability that a sum of dice is prime?

Prompted by today's Minute Math question on the MAA site (http://amc.maa.org/mathclub/5-0,problems/T-problems/T-web,ia/2005web/tb05-12-ia.shtml), I started thinking about the probability that the sum ...