Quick Intro to p-adics

What are p-adic numbers?  They are a different set of numbers first introduced by Kurt Hensel in 1897.  The motivation at that time was an attempt to bring the ideas and methods of power series methods into number theory.  They have been used in proving Fermat’s Last Theorem and have other applications in number theory.  See http://mathworld.wolfram.com/p-adicNumber.html for more information.

A little terminology needs to be introduced.  The p in p-adic represents any prime number.  For each prime, there is a new and different set of p-adic numbers.  Q2 identifies the 2-adics, Q5 represents the 5-adics, Q17 represents the 17-adics.  To keep the same notation, Q will represent the real numbers.

Another term to consider is “close.”  The basic idea is that given a number n, it is close to 0 if it is highly divisible by a prime p.  Consider the numbers 25 and 625.  Relatively speaking, 625 has more factors of 5 than does 25, or in other words, 625 has a higher divisibility by 5.  Therefore 625 is closer to 0 than 25.  This idea will be made a little bit clearer in future postings.

http://math.boisestate.edu/m287/arithmetic-on-p-adics/

8 thoughts on “Quick Intro to p-adics

  1. Samuel Coskey

    Great, I had always wondered what was the original motivation behind the p-adic numbers!

    One funny consequence of this new definition of “closeness” is the following: The numbers 25, 50, and 75 all have the same number of factors of 5 in them, so even though they are different they are all equally close to 0! This contrasts with the real numbers where two numbers that are equally close to 0 have to be the same or else negatives of one another.

    Looking forward to more!

  2. Kenny

    Does it generally hold that bigger numbers are “closer” to 0? Or like Sam said, do there exists numbers that share “closeness”?

    1. Ken Coiteux Post author

      Perhaps I may have overgeneralized a little bit. In the 5-adics, 25 is closer to 0 than 26, but that is because 26 has no factors of 5 in it. However, 25, 50, 75, and 100 all are equally close to 0 for they all contain the same powers of 5 in their factors. This will become a little clearer in the post on absolute values (http://math.boisestate.edu/m287/?p=536).

  3. Arthur Radford

    Looking at the P-adic series seems a bit difficult without the explanation of what the -adic is in the series. Does the -adic define a type of series, or function, that generates these numbers? Or is it in reference to the closeness that was discussed?

    1. Samuel Coskey

      From the OED:

      “adic”
      2. Math. With preceding symbol or numeral (esp. the generalized symbol p, denoting a prime number), forming adjectives designating numbers expressible as a sequence of digits in the base represented by the symbol (or as a power series in this quantity).

  4. Samuel Coskey

    To be close to 0, it seems to be necessary that the number be large. But this is not sufficient, for example, with $p=5$, the number 626 has no factors of 5 so it is not close to 0 at all (even though 625 is)!

  5. Marc Garland

    Me knowing nothing about p-adic numbers, this was certainly a learning experience. Also, my ignorance of these numbers also left me a little confused. It is good that you promise more clarification in your future posts. I do have the same question as many others seem to in the comments section, however. Are larger numbers always closer to zero? It is weird to think that a larger number would actually be closer to zero when in the real number system it is the opposite. Very interesting! Of course, it likely has to do with the use and function of p-adic numbers as to why they would be described this way, correct?

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