# 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:

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

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

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/

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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.

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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).

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Comment by Bruno Curfs — 2012.12.13 @ 15:06