Eikonal Blog

2010.01.05

Squares with just two different decimal digits

Filed under: mathematics, number theory, puzzles — Tags: — sandokan65 @ 16:00

I used to state here

    There exist only finitely many squares with just two decimal digits” providing a really short list

    • 38^2 = 1,444
    • 88^2 = 7,744
    • 109^2 = 11,881
    • 173^2 = 29,929
    • 212^2 = 44,944
    • 235^2 = 55,255
    • 3,114^2 = 9,696,996

    plus infinite classes:

    • 10^{2n},
    • 4\cdot 10^{2n},
    • 9\cdot 10^{2n}.

This is not correct. Thanks to Bruno Curfs for pointing this out, as well as for finding the reference [2] cited below.

  • Namely, one can see that squares of all single digit numbers satisfy targeted condition:
    • 1^2 = 01,
    • 2^2 = 04,
    • 3^2 = 09,
    • 4^2 = 16,
    • 5^2 = 25,
    • 6^2 = 36,
    • 7^2 = 49,
    • 8^2 = 64,
    • 9^2 = 81,
  • Then, there are squares of following two-digit numbers:
    • 10^2 = 100,
    • 11^2 = 121,
    • 12^2 = 144,
    • 15^2 = 225,
    • 20^2 = 400,
    • 21^2 = 441,
    • 22^2 = 484,
    • 26^2 = 676,
    • 30^2 = 900,
    • 38^2 = 1,444,
    • 88^2 = 7,744,
  • For squares of 3-digit numbers one finds:
    • 100^2 = 10,000,
    • 109^2 = 11,881
    • 173^2 = 29,929
    • 212^2 = 44,944
    • 235^2 = 55,255
    • 264^2 = 69,696 (thanks to Bruno Curfs)

Of the higher cases, there are known only infinitely long classes:

  • 10^{2n}
  • 4\cdot 10^{2n}
  • 9\cdot 10^{2n}

and following two “sporadic” occurrences:

  • 3,114^2 = 9,696,996
  • 81,619^2 = 6,661,661,161.

Author of page “1. Squares of 2 different digits” (http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math02/three01.htm) provides an implementation of the algorithm to perform exhaustive check. According to his numerical experiments, there are no other sporadic occurrences up to at least y=10^{48} (i.e. x=10^{24}).


Reference:


Similar here: More simple math wonders – https://eikonal.wordpress.com/2012/03/14/more-simple-math-wonders/ | Mental calculation of cube root of a six-digit number – https://eikonal.wordpress.com/2010/01/14/mental-calculation-of-cube-root-of-a-two-digit-number/ | Squares with just two different decimal digits – https://eikonal.wordpress.com/2010/01/05/squares-with-just-two-different-decimal-digits/ | Number theory finite concidental sums – https://eikonal.wordpress.com/2010/01/05/number-theory-finite-considental-sums/

Advertisements

4 Comments »

  1. […] | Squares with just two different decimal digits – https://eikonal.wordpress.com/2010/01/05/squares-with-just-two-different-decimal-digits/ | Number theory finite concidental sums – a Share this:StumbleUponDiggRedditLike this:LikeBe […]

    Like

    Pingback by Number theory finite concidental sums « Eikonal Blog — 2012.03.14 @ 15:30

  2. […] | Squares with just two different decimal digits – https://eikonal.wordpress.com/2010/01/05/squares-with-just-two-different-decimal-digits/ | Number theory finite concidental sums – […]

    Like

    Pingback by More simple math wonders « Eikonal Blog — 2012.03.14 @ 15:42

  3. If there is only a finite number of squares using two digits, can you give a (reference to the) complete list?
    At least one entry is missing, since 264^2 = 69696.

    Like

    Comment by Bruno Curfs — 2012.12.13 @ 15:00

  4. I found this reference http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math02/three01.htm but it does not prove that there ARE only a finite number of solutions, it just mentions the ones that are known (which are finite).

    Like

    Comment by Bruno Curfs — 2012.12.13 @ 15:06


RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: