Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For the sake of convenience, let's define an integer to be a "supersquare" if:

  • The number itself is a positive square number
  • Each digit of the number is a positive square (1, 4, 9)
  • The sum of digits is a square

For example, 144 is a supersquare because:

  • $144=12^{2}$
  • Each digit is a positive square (1, 4 and 4)
  • $1+4+4=9=3^{2}$

Here are the first few supersquares that I could find: 1, 4, 9, 144, 441, 44944

Now I'd like to ask:

  1. Are there infinite supersquares?
  2. If we allow "almost supersquares" to have exactly one 0, for example, $9941409=3153^2$, are there infinite almost supersquares?

(allowing for two 0s is trivial, because multiplying a supersquare by powers of 100 will give the required integers, e.g. 4494400, 449440000, etc.)

Unfortunately, motivation is out of my own personal curiosity and because of that I have little idea as to how to approach the problem. However, a quick check with Python for integers below $10^{14}$ has found 44944 to be the highest supersquare so far, and 4410449411449 to be the highest almost supersquare.

Edit: After searching through integers below $10^{18}$ the highest supersquare is still 44944, but here is the sequence of almost supersquares:

9941409, 141111419904, 941911011441, 1119444409444, 1144944940441, 4410449411449, 4991441999419044, 49041994144141441, 141114911949411904, ...

share|cite|improve this question
up vote 2 down vote accepted

OEIS A061269 says

Next term, if it exists, is > 90000000000 - Larry Reeves (larryr(AT), May 11 2001

Your search seems to go a bit further.

share|cite|improve this answer
Thanks for the find, I forgot to check OEIS first. There doesn't seem to be much information on it though, although the related sequences are interesting. – Sp3000 Oct 13 '11 at 12:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.