(F24 Some decimal digital problems)

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

**
81619 ^{2} = 6661661161.
**

I extended this problem as

- 3 different digits
- perfect n-th power.

For square case (b=a^{2}), I searched up to a≤10^{23} for patterns which include zero,

and up to a≤10^{25} for patterns which do not include zero by **Milos Tatarevic** (May 12, 2004).

Sometimes first solution becomes very large. For example,

2 23608 14084 16666^{2} = 5 00006 00650 66660 65606 50665 55556 (May 04, 1997)

8 81917 22853 73497^{2} = 77 77779 97990 99990 00700 07900 09009 (May 05, 1997)

9 94937 07779 87917^{2} = 98 98997 88778 79888 78977 89979 98889 (May 10, 1997)

43694 27882 45669 64251^{2} = 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 ≤ 10^{24}, b ≤ 10^{48}

678 : a ≤ 10^{25}, b ≤ 10^{50}

The solutions for higher powers are here.

I couldn't find the perfect **7-th power** consisted of 3 different digits.

- Squares of 2 different digits
- Computation techniques 1
- Possible pattern of 3 digits
- Computation techniques 2, results and higher power cases

[1] Richard K. Guy, *Unsolved Problems in Number Theory (Second Edition)*, Springer, 1994.

[2] Ilan Vardi, *Computational Recreations in Mathematica*, Addison-Wesley.

Chapter 1 4/n=1/a+1/b+1/c |
"Mathematician's Secret Room" | Chapter 3 n=(x+y+z)(1/x+1/y+1/z) |
---|---|---|

Chapter 1 (Japanese) | index (Japanese) | Chapter 3 (Japanese) |

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