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