A018884 Squares using at most two distinct digits, not ending in 0.

1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 225, 441, 484, 676, 1444, 7744, 11881, 29929, 44944, 55225, 69696, 9696996, 6661661161

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base
• more
• hard

Comments

No other terms below 10^41.

The sequence is probably finite.

The two distinct digits of a term cannot both be in the set {0,2,3,7,8}. Looking at the digits (with leading zeros) of i^2 mod 10^4 for 0 <= i < 10^4 shows that there are no repunit terms > 10 and the two distinct digits of a term must be one of the following 21 pairs: '01', '04', '09', '12', '14', '16', '18', '24', '25', '29', '34', '36', '45', '46', '47', '48', '49', '56', '67', '69', '89'. - Chai Wah Wu, Apr 06 2019

References

• Richard K. Guy, Unsolved Problems in Number Theory, Section F24 (at p. 262) (Springer-Verlag, 2d ed. 1994).

Links

• Eric Weisstein's World of Mathematics, Square Number.

Crossrefs

Cf. A000290, A016069, A016070, A018885.

Programs

Mathematica

Flatten[Table[Select[Flatten[Table[FromDigits/@Tuples[{a,b},n],{n,10}]], IntegerQ[ Sqrt[#]]&],{a,9},{b,9}]]//Union (* Harvey P. Dale, Sep 21 2018 *)