Problems & Puzzles: Puzzles

Puzzle 643. 134757431 134757431 is a palprime that can be expressed in three distinct ways using twice the nine integers from 1 to 9, one as bases and another as powers as shown below: 134757431 = 1^7+2^3+3^8+4^5+5^4+6^2+7^1+8^9+9^6 =                 = 1^7+2^5+3^8+4^1+5^2+6^4+7^3+8^9+9^6 =                 = 1^7+2^8+3^4+4^2+5^3+6^5+7^1+8^9+9^6 Q. Can you find another palprime expressible in more than 3 distinct ways?

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Contributions came from J. K. Andersen, Jan van Delden, Ryan Bailey, Gaurav Kamur, Hakan Summakoglu, Emmanuel Vantieghem and Farideh & Jahangeer.

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Andersen wrote:

134757431 is the only palindrome at all which can be expressed in 3 ways.

There are two non-palindromic primes with 5 expressions:
5053579 = 1^5+2^6+3^8+4^9+5^2+6^4+7^1+8^3+9^7
      = 1^5+2^9+3^8+4^3+5^2+6^4+7^1+8^6+9^7
      = 1^8+2^6+3^4+4^9+5^2+6^5+7^1+8^3+9^7
      = 1^8+2^9+3^4+4^3+5^2+6^5+7^1+8^6+9^7
      = 1^9+2^8+3^1+4^4+5^3+6^5+7^2+8^6+9^7

42458831 = 1^5+2^6+3^8+4^2+5^3+6^4+7^9+8^7+9^1
       = 1^5+2^6+3^8+4^3+5^1+6^4+7^9+8^7+9^2
       = 1^6+2^8+3^2+4^5+5^1+6^3+7^9+8^7+9^4
       = 1^8+2^6+3^4+4^2+5^3+6^5+7^9+8^7+9^1
       = 1^8+2^6+3^4+4^3+5^1+6^5+7^9+8^7+9^2

There is one integer, neither prime nor palindrome, with 6 expressions:
1094207 = 1^5+2^6+3^8+4^9+5^1+6^4+7^7+8^3+9^2
      = 1^5+2^9+3^8+4^2+5^3+6^4+7^7+8^6+9^1
      = 1^5+2^9+3^8+4^3+5^1+6^4+7^7+8^6+9^2
      = 1^8+2^6+3^4+4^9+5^1+6^5+7^7+8^3+9^2
      = 1^8+2^9+3^4+4^2+5^3+6^5+7^7+8^6+9^1
      = 1^8+2^9+3^4+4^3+5^1+6^5+7^7+8^6+9^2

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Jan wrote:

The given palprime is the only solution which can be written in this fashion using three different representations. More than 3 is not possible.

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Ryan wrote:

Q. No, Here is a list of all Palprimes expressible in some format of the defined parameters in the following order:

 

PalPrime, # of ways expressible, way(s) of being expressed

30103 1 1^9 + 2^8 + 3^7 + 4^6 + 5^3 + 6^1 + 7^5 + 8^2 + 9^4
32323 1 1^7 + 2^9 + 3^8 + 4^6 + 5^2 + 6^3 + 7^5 + 8^4 + 9^1
35353 1 1^7 + 2^9 + 3^8 + 4^1 + 5^6 + 6^5 + 7^2 + 8^4 + 9^3
36563 1 1^8 + 2^7 + 3^9 + 4^6 + 5^1 + 6^5 + 7^2 + 8^4 + 9^3
70607 1 1^9 + 2^7 + 3^8 + 4^4 + 5^3 + 6^6 + 7^5 + 8^2 + 9^1
74747 1 1^7 + 2^8 + 3^9 + 4^5 + 5^1 + 6^6 + 7^2 + 8^3 + 9^4
76367 1 1^9 + 2^7 + 3^6 + 4^8 + 5^5 + 6^3 + 7^1 + 8^2 + 9^4
79697 1 1^9 + 2^1 + 3^8 + 4^7 + 5^5 + 6^6 + 7^3 + 8^2 + 9^4
90709 1 1^5 + 2^9 + 3^7 + 4^8 + 5^6 + 6^3 + 7^1 + 8^2 + 9^4
93139 1 1^9 + 2^8 + 3^2 + 4^7 + 5^6 + 6^4 + 7^1 + 8^3 + 9^5
94349 1 1^6 + 2^7 + 3^9 + 4^8 + 5^4 + 6^5 + 7^1 + 8^3 + 9^2
96269 1 1^5 + 2^9 + 3^8 + 4^6 + 5^7 + 6^1 + 7^3 + 8^2 + 9^4
96469 1 1^2 + 2^9 + 3^7 + 4^8 + 5^6 + 6^5 + 7^1 + 8^4 + 9^3
97379 1 1^9 + 2^8 + 3^1 + 4^6 + 5^7 + 6^5 + 7^2 + 8^3 + 9^4
98689 1 1^8 + 2^7 + 3^9 + 4^3 + 5^6 + 6^2 + 7^1 + 8^4 + 9^5
1120211 1 1^4 + 2^8 + 3^6 + 4^9 + 5^1 + 6^2 + 7^7 + 8^5 + 9^3
1160611 1 1^8 + 2^4 + 3^2 + 4^9 + 5^6 + 6^3 + 7^7 + 8^1 + 9^5
1177711 1 1^5 + 2^1 + 3^9 + 4^8 + 5^2 + 6^3 + 7^7 + 8^6 + 9^4
1180811 1 1^2 + 2^1 + 3^9 + 4^8 + 5^5 + 6^3 + 7^7 + 8^6 + 9^4
1186811 1 1^5 + 2^7 + 3^3 + 4^9 + 5^8 + 6^2 + 7^4 + 8^1 + 9^6
1363631 1 1^9 + 2^8 + 3^4 + 4^2 + 5^1 + 6^5 + 7^7 + 8^3 + 9^6
1556551 1 1^1 + 2^2 + 3^4 + 4^9 + 5^8 + 6^6 + 7^7 + 8^5 + 9^3
1688861 1 1^7 + 2^9 + 3^6 + 4^1 + 5^5 + 6^8 + 7^2 + 8^4 + 9^3
1712171 1 1^9 + 2^5 + 3^4 + 4^7 + 5^6 + 6^8 + 7^3 + 8^1 + 9^2
1755571 1 1^9 + 2^6 + 3^4 + 4^7 + 5^2 + 6^8 + 7^3 + 8^1 + 9^5
1805081 1 1^7 + 2^9 + 3^2 + 4^3 + 5^5 + 6^8 + 7^6 + 8^4 + 9^1
1876781 1 1^5 + 2^9 + 3^4 + 4^1 + 5^7 + 6^8 + 7^6 + 8^2 + 9^3
1976791 1 1^8 + 2^6 + 3^5 + 4^7 + 5^9 + 6^1 + 7^3 + 8^2 + 9^4
1982891 1 1^8 + 2^1 + 3^6 + 4^7 + 5^9 + 6^5 + 7^2 + 8^4 + 9^3
1995991 1 1^2 + 2^1 + 3^9 + 4^7 + 5^4 + 6^8 + 7^5 + 8^6 + 9^3
7884887 2 1^5 + 2^9 + 3^1 + 4^2 + 5^6 + 6^3 + 7^8 + 8^7 + 9^4 AND
 1^6 + 2^4 + 3^9 + 4^3 + 5^5 + 6^2 + 7^8 + 8^7 + 9^1
7891987 1 1^9 + 2^2 + 3^1 + 4^3 + 5^6 + 6^5 + 7^8 + 8^7 + 9^4
7996997 1 1^4 + 2^1 + 3^5 + 4^7 + 5^9 + 6^3 + 7^8 + 8^6 + 9^2
134757431 3 1^7 + 2^3 + 3^8 + 4^5 + 5^4 + 6^2 + 7^1 + 8^9 + 9^6 AND
 1^7 + 2^5 + 3^8 + 4^1 + 5^2 + 6^4 + 7^3 + 8^9 + 9^6 AND
 1^7 + 2^8 + 3^4 + 4^2 + 5^3 + 6^5 + 7^1 + 8^9 + 9^6
135101531 1 1^8 + 2^2 + 3^6 + 4^4 + 5^1 + 6^3 + 7^7 + 8^9 + 9^5
135929531 1 1^5 + 2^3 + 3^4 + 4^7 + 5^6 + 6^8 + 7^1 + 8^9 + 9^2
136737631 1 1^3 + 2^2 + 3^4 + 4^5 + 5^6 + 6^8 + 7^7 + 8^9 + 9^1
387434783 1 1^6 + 2^7 + 3^8 + 4^1 + 5^5 + 6^2 + 7^3 + 8^4 + 9^9

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Gaurav wrote:

There are numbers expressible in more than 3 ways (using twice the nine integers from 1 to 9, one as bases and another as powers) but none of them are palprime.

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Hakan wrote:

There are only 38 palprimes (41 expressions) like this.
There are only 2 palprimes in more than 1 distinct ways.

(2 ways) 7884887 = 1^6+2^4+3^9+4^3+5^5+6^2+7^8+8^7+9^1
                          = 1^5+2^9+3^1+4^2+5^6+6^3+7^8+8^7+9^4

(3 ways) 134757431 = 1^7+2^3+3^8+4^5+5^4+6^2+7^1+8^9+9^6
                              = 1^7+2^5+3^8+4^1+5^2+6^4+7^3+8^9+9^6
                              = 1^7+2^8+3^4+4^2+5^3+6^5+7^1+8^9+9^6

There isn't any other palprime expression in more than 3 distinct ways. (I added all expressions in additional file)

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Emmanuel wrote:

There are no more palprimes that have more than three representations.

If we use the ten numbers  0, 1, 2, ..., 8, 9  as bases and as exponents we must agree how to interprete the expression  0^0.

I choosed to 'drop' it, what's the same as setting it equal to zero.

Then, the maximum number of representations of palprimes is 8 and is reached by  70607  and  134787431 :

 70607 = 0^9+1^0+2^7+3^8+4^4+5^3+6^6+7^5+8^2+9^1

           = 0^9+1^3+2^8+3^0+4^7+5^5+6^6+7^1+8^4+9^2

           = 0^9+1^7+2^5+3^6+4^8+5^3+6^1+7^0+8^4+9^2

           = 0^7+1^4+2^9+3^8+4^6+5^0+6^2+7^3+8^1+9^5

           = 0^7+1^9+2^5+3^6+4^8+5^3+6^1+7^0+8^4+9^2

           = 0^4+1^7+2^9+3^8+4^6+5^0+6^2+7^3+8^1+9^5

           = 0^3+1^9+2^8+3^0+4^7+5^5+6^6+7^1+8^4+9^2

                    = 1^9+2^7+3^8+4^4+5^3+6^6+7^5+8^2+9^1

1234757431 = 0^8+1^4+2^7+3^2+4^1+5^0+6^5+7^3+8^9+9^6

                   = 0^7+1^0+2^3+3^8+4^5+5^4+6^2+7^1+8^9+9^6

                   = 0^7+1^0+2^5+3^8+4^1+5^2+6^4+7^3+8^9+9^6

                   = 0^7+1^0+2^8+3^4+4^2+5^3+6^5+7^1+8^9+9^6

                   = 0^4+1^8+2^7+3^2+4^1+5^0+6^5+7^3+8^9+9^6

                           = 1^7+2^3+3^8+4^5+5^4+6^2+7^1+8^9+9^6

                           = 1^7+2^5+3^8+4^1+5^2+6^4+7^3+8^9+9^6

                           = 1^7+2^8+3^4+4^2+5^3+6^5+7^1+8^9+9^6

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Farideh & Jahangeer wrote:

There is no palprime expressible in more than 3 distinct ways. In fact 134757431 is the only palprime with more than two representations of the mentioned form.

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