Problems & Puzzles: Puzzles

Puzzle 380. A follow up to Puzzle 42

Anton Vrba recently contributed to the Puzzle 42, and added the following nice puzzle related to the same issue (cryptarithmetic):

If TWO is even and THREE,  FIVE, THIRTEEN  are  prime.  

Then all the following should also be prime

  1. TWO * THREE + FIVE 
  2. TWO * THREE + THIRTEEN    
  3. TWO * FIVE + THREE
  4. TWO * FIVE + THIRTEEN
  5. TWO * THIRTEEN + THREE
  6. TWO * THIRTEEN + FIVE
  7. THREE * FIVE + TWO
  8. THREE * THIRTEEN + TWO
  9. FIVE * THIRTEEN + TWO

Questions.

1)   Find the minimum solution for each of the nine equations individually (you may use any base)

2)  Find a simultaneous solution for any two, three, four … and possibly all nine equations.
 

Contributions came from: Fred Schneider, Jacques Tramu, Charles Greathouse & Farideh Firoozbakht.

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Fred wrote:

I used a brute force search in PARI to find all the solutions:

Minimum solution by equation number:
1: 1323131 for primes: TWO=126 and THREE=10433 and THIRTEEN=10541339 and FIVE=8573
2: 12379837 for primes:TWO=156 and THREE=10499 and THIRTEEN=10741993 and FIVE=2789
3: 445583 for primes: TWO=156 and THREE=10499 and THIRTEEN=10741993 and FIVE=2789
4: 11177077 for primes: TWO=156 and THREE=10499 and THIRTEEN=10741993 and FIVE=2789
5: 1515871211 for primes: TWO=102 and THREE=14633 and THIRTEEN=14861339 and FIVE=7853
6: 1366016567 for primes: TWO=126 and THREE=10433 and THIRTEEN=10841339 and FIVE=7853
7: 29281867 for primes: TWO=156 and THREE=10499 and THIRTEEN=10741993 and FIVE=2789
8: 109977789949 for primes: TWO=162 and THREE=10433 and THIRTEEN=10541339 and FIVE=7583
9: 24707030989 for primes: TWO=156 and THREE=10499 and THIRTEEN=10341997 and FIVE=2389


Number of solutions by count:
n   Given the constraints of TWO, THREE THIRTEEN and FIVE, number of cases where n of the 9 equations are primes
0 : 1013
1 : 709
2 : 350
3 : 128
4 : 36
5 : 4
6 : 1
7 : 0
8 : 0
9 : 0

The winner was:
**** Multi-solution found = 6 for primes: TWO=156 and THREE=10499 and THIRTEEN=10741993 and FIVE=2789 for eqs: 1 2 3 4 5 7
(Note: This quartet generated 4 of the minimum solutions above)


Other multi-solution examples (including all four "5"s):
**** Multi-solution found = 2 for primes: TWO=102 and THREE=14633 and THIRTEEN=14861339 and FIVE=7853 for eqs: 1 5
**** Multi-solution found = 3 for primes: TWO=102 and THREE=14699 and THIRTEEN=14861993 and FIVE=5879 for eqs: 1 7 9
**** Multi-solution found = 4 for primes: TWO=150 and THREE=13877 and THIRTEEN=13481779 and FIVE=6427 for eqs: 2 3 4 8
**** Multi-solution found = 5 for primes: TWO=180 and THREE=15233 and THIRTEEN=15421337 and FIVE=9463 for eqs: 1 3 6 7 8
**** Multi-solution found = 5 for primes: TWO=350 and THREE=31477 and THIRTEEN=31643779 and FIVE=2687 for eqs: 1 4 6 7 9
**** Multi-solution found = 5 for primes: TWO=498 and THREE=45077 and THIRTEEN=45204773 and FIVE=6217 for eqs: 2 3 4 5 7
**** Multi-solution found = 5 for primes: TWO=498 and THREE=45077 and THIRTEEN=45204773 and FIVE=6217 for eqs: 2 3 4 5 7

Counts by equation:
1 : 258
2 : 268
3 : 266
4 : 257
5 : 196
6 : 171
7 : 219
8 : 152
9 : 176
 

***

Jacques wrote:
BASE 10 : Solution for six equations :
 
TWO = 156
THREE = 10499
FIVE = 2789
THIRTEEN = 10741993
 
156 * 10499 + 2789  =    1640633
156 * 10499 + 10741993  =   12379837
156 * 2789 + 10499   =  445583
156 * 2789 + 10741993  =    11177077
156 * 10741993 + 10499   =   1675761407
10499 * 2789 + 156    =  29281867

***

Charles wrote:

...here are (I believe) the
correct answers for question 1, or at least the first few expressions.
The format is (TWO, THREE, FIVE, THIRTEEN): PrimeAnswer, where
PrimeAnswer is the minimum prime value of the question's expression.

1. (126, 10433, 8573, 10541339): 1,323,131
2. (156, 10499, 2789, 10741993): 12,379,837
3. (156, 10499, 2789, 10741993): 445,583
4. (156, 10499, 2789, 10741993): 11,177,077
5. (102, 14633, 7853, 14861339): 1,515,871,211
6. (126, 10433, 7853, 10841339): 1,366,016,567
7. (156, 10499, 2789, 10741993): 29,281,867

I solved the problem with the help of a computer program I wrote to
quickly form the numbers and check for primality.
 

***

Farideh wrote:
1. What is the meaning of "minimum solution" ?
  
2. T > 0  or  T >= 0 (F > 0  or  F >= 0)? 
For example If T can be zero,  (078, 03499, 6529, 03540991) is a solution for the
first four equations and is also a solution for the 6-th equation.
 
I found all of the solutions (TWO, THREE, FIVE, THIRTEEN)  of the first equation 
" TWO*THREE + FIVE = prime ".
 
Three of them are:
 
111110100 :  (156, 10499, 2789, 10741993 )
This solution is a solution for the first five equations and also is a solution for the
7-th equation (for this reason I named it 111110100).
 
111000011 : (120, 16477, 5387, 16341779)
This solution is a solution for the first three equations and also is a solution for the
last two equations.
 
101001110 : (180, 15233, 9463, 15421337)
This solution is a solution for the first, 3-th, 6-th, 7-th & 8-th equations.

***

Bernardo Boncompagni wrote (Dec. 23, 06):

The smallest base for which all constraints hold is 19. In base 19, with
the usual notation 1,2,...,9,A,... we have:

TWO = 1BC = 582 decimal
THREE = 12E44 = 149173 decimal
FIVE = 9A84 = 65497 decimal
THIRTEEN = 12AE144D = 1014557377 decimal

If we remove the constraint that every letter should represent a
distinct number, we have a solution in base 10:

TWO = 210
THREE = 25633
FIVE = 1103
THIRTEEN = 25162339

Without the distinct digit constraint, it is amazing that no smaller
base holds a solution, as we can solve 8 out of 9 equations in base 3
already:

TWO = 101 = 10 decimal
THREE = 12211 = 157 decimal
FIVE = 1011 = 31 decimal
THIRTEEN = 12021111 = 3847 decimal

where only the 5th equation holds a composite term.

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