Problems & Puzzles: Puzzles

 Puzzle 313. Squares having only k distinct digits   To an old friend of mine and my prime pages: Enoch Haga.   Case k=2 For sure you already know that the largest known (May 2005) number  -trivial patterns apart (*) - such that its square has only 2 distinct digits is: 81619 816192 = 6661661161 Surprisingly this number 81619 is also a prime number! Case k=3 The largest known number -patterns apart- such that its square has 3 distinct digits is: 81401637345465395512991484 (26 digits) 814016373454653955129914842 = 6626226562522666562566262626266252566552622656522256...(**) The largest known prime number -patterns apart- such that its square has 3 distinct digits is: 999997323321167445187 (prime, 21 digits) .........(***) 9999973233211674451872 = 999994646649499499946646996649944649464969 For the case of squares with 3 distinct digits, the patterns are not totally trivial. Hisanori Mishima has also obtained (all?) the possible patterns (31) three of which may produce provable primes whose squares has 3 distinct digits: 1) 1(0)n3(0)n1 --> square digits:0,1,6 2) 1(0)n8(0)n1 --> square digits:0,1,6 3) 9.10^n+1 --> square digits:0,1,8 As a matter of fact, the records for each mode are: For the model 1) Heuer found a palprime with n = 34940 For the model 2) Heuer found a palprime with n = 37814 For the model 3) Benson found a (GF!) prime with n = 70500 Questions: 1. Can you get a prime larger than 999997323321167445187 whose square has 3 distinct digits? 2. Can you get a prime larger than the already have been found for the models 1), 2) & 3) above? ________ (*) Three possible patterns are a(0)n, for a=1, 2 & 3 (**) This is the greatest square found in the Hisanori & Milos Tatarevic's database for k=3. (***) I found this prime analyzing the primality the numbers in the Hisanori & Milos Tatarevic's database for k=3.

Faride Firoozbakht has found a new pattern outside of the 31 ones obtained by Hisanori Mishima (this has justified my skeptical parentheses - all? - above).  Faride has also given some detail inside the Pattern #1 of the Hisanori's ones.

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Faride Firoozbakht wrote:

Squares of numbers of the form 10^m+10^n+11 where 2 < n < m- 1 & n#[(m+1)/2] has three distinct digits (0,1,2).

Example : (10^9+10^4+11)^2=1000020022100220121

(10^9+10^5+11)^2=1000200032002200121 note that 5=[(9+1)/2])

primes (or probable primes) of such form :

p1=10^1000+10^360+11
p2=10^2000+10^1280+11
p3=10^3000+10^1750+11

Also there exist many patterns which may produce provable primes whose squares has 3 distinct digits(0,1,2), namely numbers of the form 10^m1+10^m2+...+10^mk where 3 doesn't divide k and with some conditions on mi's. If k=4 & m3=m4+1=1 we get the above case.

...

I also found the following new pattern:

If 0 < m < n-1
f(m,n) = 5.0(m).5.0(n).2.0(n+2).5.0(m).5
g(m,n) = 25.0(m-1).5.0(m).25.0(n-m-2).2.0(m).2.0(n-m+1).
5.0(m-1).5.0(m+1).5.0(n-m-1).2.0(m).2.0(n-m+1).
25.0(m-1).5.0(m).25
then f(m,n)^2=g(m,n)

Some special cases:

m=1
f(1,3) = 505000200000505
g(1,3) = 255025202000550050202000255025

f(1,4)= 50500002000000505
g(1,4)= 2550250202000055005002020000255025
.
.
************************************************************
m=2
f(2,4) = 5005000020000005005
g(2,4) = 25050025200200050500050200200025050025

f(2,5) = 500500000200000005005
g(2,5) = 250500250200200005050005002002000025050025
.
.
************************************************************
m=3
f(3,5) = 50005000002000000050005
g(3,5) = 2500500025200020005005000050200020002500500025

f(3,6) = 5000500000020000000050005
g(3,6) = 25005000250200020000500500005002000200002500500025

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