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All of us know the extensive application of integer factorization in Cryptography. But, I would like to know its applications in other fields.

If anyone knows some applications of integer factorization with applications in data science or data mining. Please state them as well.

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  • $\begingroup$ I meant the areas where factorization plays an important role in solving the problems. $\endgroup$ – Mayank Aug 8 '16 at 13:06
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    $\begingroup$ In group theory, the prime factorization plays an important role. For example, the number of groups of order $n$ upto isomorphy heavily depends on the prime factorization of $n$. $\endgroup$ – Peter Aug 8 '16 at 13:08
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    $\begingroup$ The prime factorization can also be useful to find divisibility-rules. For example, when is a number divisible by $7$ ? $\endgroup$ – Peter Aug 8 '16 at 13:11
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    $\begingroup$ I use integer factorization whenever I simplify fractions. $\endgroup$ – Joel Reyes Noche Aug 8 '16 at 13:34
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    $\begingroup$ The primality (non-amenability to factorisation) of numbers can be significant in biology. See en.wikipedia.org/wiki/Periodical_cicadas. $\endgroup$ – Mr. Chip Aug 8 '16 at 13:46
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The field of Combinatorics.


For example:

Find the number of positive integer divisors of $48$.


Using the prime factorization of $48=2^4\cdot3^1$, we can easily answer this question:

  • In each divisor, the factor $2$ can appear between $0$ and $4$ times, i.e., $5$ different combinations
  • In each divisor, the factor $3$ can appear between $0$ and $1$ times, i.e., $2$ different combinations

Hence there are $5\cdot2=10$ divisors:

  • $2^0\cdot3^0$
  • $2^0\cdot3^1$
  • $2^1\cdot3^0$
  • $2^1\cdot3^1$
  • $2^2\cdot3^0$
  • $2^2\cdot3^1$
  • $2^3\cdot3^0$
  • $2^3\cdot3^1$
  • $2^4\cdot3^0$
  • $2^4\cdot3^1$
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