Question:
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
Approach:
Every integer greater than one has a unique prime factorization. This means that each integer greater than one can be represented as a unique product of prime numbers. For example, the number 5,784 can be written as:
\[2^3\times 241\]
Where 2 and 241 are prime. Likewise, the number 54,887,224 can be written as:
\[2^3\times 7 \times 53 \times 18493\]
Where 2, 7, 53, and 18493 are prime. For this problem, our goal is to find the largest prime factor of the number 600,851,475,143. An interesting thing to note is that the mathematician F. N. Cole, back in 1903, found the prime factorization of the number 147,573,952,589,676,412,927, which is 193,707,721 × 761,838,257,287. It took him three years worth of effort to do so. While our number is not quite as large as Cole’s, such problems today are trivial. Computer algebra systems such as Mathematica can factorize large numbers very quickly. What took Cole 3 years back in 1903 now takes less than a second today.
We have some discretion in our approach to solving this problem. You can probably use Mathematica to solve it with a single line of code. On the other hand, if you’re using a bare programming language like C++, you would probably need to write more code. Your choice on what to do depends on what you want to learn from solving the problem. I had previously solved this problem using VBA which first involved creating a list of prime numbers, using the list to find the prime factors of 600,851,475,143, and then selecting the largest one as my answer. With R, I’ve taken a middle approach, compromising between arriving at an immediate answer with a computer algebra system and having to create a list of primes from scratch. In this case, I used a list of primes I found off a website and saved them into a file called primes.csv. My mental outline is as follows:
1. Obtain a list of prime numbers
2. Find which of these prime numbers divide 600,851,475,143 evenly.
3. Of the primes that divide 600,851,475,143 evenly, find the largest one.
Solution:
|
1 2 |
primes <- sort(as.vector(as.matrix(read.csv("primes.csv",header=FALSE)))) max(primes[which(600851475143 %% primes == 0)]) |
The first line does five things. First, it imports the list of prime numbers as a data frame, then converts the data frame into a matrix, then converts the matrix into a vector, and then sorts the primes within the vector in ascending order. This vector is then assigned to the variable primes. The second line accomplishes three things. The which() function returns the indices of the elements that divide 600,851,475,143 evenly. These indices are then used to extract the corresponding elements of the vector primes. Finally, the function max() returns the largest of these primes, giving us the answer, 6857.
I thought this answer would avoid the tedium of having to calculate each prime myself, but was not so trivial that I could find the answer without learning anything. I thought this method was a good way to demonstrate the use of indexing to find the solution.