# Tagged Questions

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### Largest prime factor of a number

In Project Euler problem 3, where we have to find the largest prime factor of a number, one of the solution i came across is ...
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### Any simpler way to do Pollard's p-1 method?

I found calculating factorization by Pollard's p-1 method is almost impossible if use a conventional scientific calculator. For example, I am trying to factor ...
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### Probability Distribution of Count of Factors for All Numbers

Is the following a known thing? Define "factor count" as the count of factors each number has, then subtract 1. Ignore the number "1" as a factor. For example: Prime numbers have a factor count ...
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### Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
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### Unique number of numbers multiplied together

I'm sure this has been asked before, but how many unique numbers can be made from multiplying $4$ numbers, each between $1$ and $100$? My guess is the all numbers from $1$ to $100^4$ except those ...
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### Probability distribution of the (size of the) smallest prime factor

Related question: Expected smallest prime factor Background: Given a toolbox of factorization algorithms (like trial division, ECM, quadratic sieve, GNFS) and a set of large composite numbers, I'm ...
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### Prime factorization: easiest way?

For prime factorization, is there another way of doing it, distinct from dividing the number by a series of primes (starting by the smallest)? Couldn't we also pick the same series of primes and ...
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### Intermediate Problem Solving Patterns involving Prime Factoring

a and b are positive integers such that $a\times b= 500000000,$ where neither a nor b contain any zeros. Find a and b where $a<b.$
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### If n > 3 and (n + 1) is a square, is there any n that is a prime?

I am looking at properties of squares and came about this property. I am investigating the difference of squares in relation to primes.
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### Factoring a number $p^a q^b$ knowing its totient

We are given: $n=p^aq^b$ and $\phi(n)$, where $p,q$ are prime numbers. I have to calculate the $a,b,p,q$, possibly using computer for some calculations, but the method is supposed to be symbolically ...
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### How to find the factors of numbers around 1e7?

I don't have a maths background but I'm solving problems on the awesome Project Euler .net in JavaScript as programming practice. I don't want to link directly to the question or post it verbatim ...
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### What are the specifics and the possible outputs of Pollard's Rho algorithm?

I'm trying to implement a simple prime factorization program (for Project Euler), and want to be able to use Pollard's Rho algorithm. I read the Wikipedia, wolfram|alpha, and planet math explanations ...
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### Could G. H. Hardy make a product of two primes so big he couldn't find out which?

This question of course began as a slightly irreverent play on the riddle, "Can God make a stone so big He can't lift it?" Then I wondered about the answer. Is it possible to exhibit a number that is ...
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### How can we prove that among positive integers any number can have only one prime factorization?

I have read right from school that prime factorization is unique, but have never found proof for this. Can someone show me the proof?
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### Interesting prime factorization function divisibility problem [duplicate]

Possible Duplicate: Is the set of all numbers which divide a specific function of their prime factors, infinite? Let the function $f(n) =(p_1^{a+1}-1)(p_2^{b+1}-1)...$ where $n$ is an ...
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### Is there a name for a number whose factors' exponents are all prime?

For instance, 864, whose factorization is 2^5 x 3^3.
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### Unique factorization less than 100

How do I approach this problem using unique factorization?... How many numbers are product of (exactly) $3$ distinct primes $< 100$? edit: Just to add to that, How does unique factorization ...
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### lower bounds for maximum computing times for integer factorisation

Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...
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### Factoring short intervals

There are algorithms (e.g., SIQS) that factor individual numbers. For large ranges of numbers, sieving is more efficient: for example, $(x^2,x^2+x)$ can be factored in time roughly linear in $x$. ...
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### How to determine in polynomial time if a number is a product of two consecutive primes?

How to determine in polynomial time if a number is a product of two consecutive primes? All I can figure out is that if Cramér's conjecture is true, then we can use the AKS primality test to find ...
Suppose you are given a number $n$ and told that the sum of its prime factors is $s$. I'm looking for an efficient algorithm that checks the truth of the statement. Obviously one can simply ...