## Chapter 2. Squares consisted of 3 different digits (F24 Some decimal digital problems)

(October 31, 2008) [Japanese]

### Abstract

In Unsolved Problem in Number Theory F 24, Japanese mathematician Shin Hitotsumatsu asked the proof or the contradiction that
are there perfect square number consisted of only 2 different digits and
non-trivial patterns ("trivial" means 100..00, 400..00, 900..00) or not.

The largest known solution is

816192 = 6661661161.

I extended this problem as

1. 3 different digits
2. perfect n-th power.

For square case (b=a2), I searched up to a≤1023 for patterns which include zero,
and up to a≤1025 for patterns which do not include zero by Milos Tatarevic (May 12, 2004).

Sometimes first solution becomes very large. For example,

2 23608 14084 166662 = 5 00006 00650 66660 65606 50665 55556 (May 04, 1997)
8 81917 22853 734972 = 77 77779 97990 99990 00700 07900 09009 (May 05, 1997)
9 94937 07779 879172 = 98 98997 88778 79888 78977 89979 98889 (May 10, 1997)
43694 27882 45669 642512 = 19091 90001 99900 10111 09190 09010 99119 91001 (May 06, 1998)

And I couldn't find the perfect square number consisted of 013 and 689 under,

013 : a ≤ 1024, b ≤ 1048
678 : a ≤ 1025, b ≤ 1050

The solutions for higher powers are here.
I couldn't find the perfect 7-th power consisted of 3 different digits.

### References

 Richard K. Guy, Unsolved Problems in Number Theory (Second Edition), Springer, 1994.
 Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley.

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima