Problems & Puzzles: Puzzles

Puzzle 564.- Primorials & primes

JM Bergot sent the following puzzle:

Take a primorial and replace succesively each '*' with a '+', one at the time.

An example for 13#  2 + 3*5*7*11*13=15017 (prime)

2*3 + 5*7*11*13 = 5011 (prime)

2*3*5 + 7*11*13 = 1031 (prime)

2*3*5*7 + 11*13=353 (prime)

2*3*5*7*11 + 13 = 2323 (fail)

Q1. Will each primorial produce at least one prime in this way?

Q2. What is the largest primorial you can find producing not one prime?

Q3. What primorial can you find producing the largest number of primes?

Contributions came from: Torbjörn Alm, Emmanuel Vantieghem, Jeff Heleen, Antoine Verroken, Jan van Delden, Farid Lian (BTW, someone has given incorrect answers)

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Torbjörn wrote:

I have checked the first 500 primorials.

The example is outstanding!

Size distribution
0  482
1  12
2  1
3  0
4  1

37# gives 2 primes.
37#
2+3*5*7*11*13*17*19*23*29*31*37 = +3710369067407(prime)
2*3+5*7*11*13*17*19*23*29*31*37 = +1236789689141(prime)
2*3*5+7*11*13*17*19*23*29*31*37 = +247357937857(fail)
Size=2

2273# was the highest non-zero size

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Emmanuel Vantieghem wrote:

For most values of  n (>2), at least one of the sums  p_k#+(p_n#)/(p_k#)  (k = 1, 2, ...,n-1) is prime.  The exceptions are  n =35, 40, 70, 150, 153, 160, 164, 165, 200, 248, 275, 298, 300, 308, 319, 339, 376, 413, 435, 447, 453, 497, 500, 524, 538, 550, 555, 558, 572, 618, 621, 623, 629, 649, 663  and the next one is > 800.
However, the primes I met are not all proved primes.
The first time 2 primes are among those sums occurs for  n = 3
The first time 3 primes are among those sums occurs for  n = 4
The first time 4 primes are among those sums occurs for  n = 6
The first time 5 primes are among those sums occurs for  n = 12
The first time 6 primes are among those sums occurs for  n = 15
The first time 7 primes are among those sums occurs for  n = 119
The first time 8 primes are among those sums occurs for  n = 138
Also, the last two results deal with probable primes.  I did not find an  n < 200  that gave 9 (probable) primes.

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Jeff Heleen wrote:

Q1. No.
Q2. 149#, 173#, 349# produce no primes.
Q3. 47# and 269# each produce 6 primes.

3#, 5#, 7# produce all primes.

All other primorials <=353# produce only 1 to 5 primes each.

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Antoine Verroken wrote:

primes from 2 to 547 ( 100th prime )

Q1. : no ; # 149 , # 173 , # 349 do not produce one prime

Q2. : # 349

Q3. : largest number of primes : 6 in # 47 , # 269 , # 463

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Jan van Delden wrote:

Q2:   The largest primorial with no primes of the indicated form, (I had time for to find): 4957#. (The smallest is 149#)
Q3:   Changing the first k consecutive ‘*’-s into ‘+’-s: No larger solutions than k=4, 13#.
The largest number of prime-terms: 10, for 2129#.

Related: all splits are prime: only 3#, 5# and 7#.

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Farid Lian wrote:

Q1. Will each primorial produce at least one prime in this way?

No, please to see the first 17 primorials, … 11#, 17#, 19#, 23#, 41#, 43# … has not primes…

2,1,5

primorial  -------------> 3# has 1 prime(s)

3,1,17

3,2,11

primorial  -------------> 5# has 2 prime(s)

4,1,107

4,2,41

4,3,37

primorial  -------------> 7# has 3 prime(s)

primorial  -------------> 11# has 0 prime(s)

6,1,15017

6,2,5011

6,3,1031

6,4,353

primorial  -------------> 13# has 4 prime(s)

primorial  -------------> 17# has 0 prime(s)

primorial  -------------> 19# has 0 prime(s)

primorial  -------------> 23# has 0 prime(s)

10,1,3234846617

primorial  -------------> 29# has 1 prime(s)

11,1,100280245067

primorial  -------------> 31# has 1 prime(s)

12,1,3710369067407

12,2,1236789689141

primorial  -------------> 37# has 2 prime(s)

primorial  -------------> 41# has 0 prime(s)

primorial  -------------> 43# has 0 prime(s)

15,1,307444891294245707

primorial  -------------> 47# has 1 prime(s)

primorial  -------------> 53# has 0 prime(s)

17,1,961380175077106319537

primorial  -------------> 59# has 1 prime(s)

primorial  -------------> 61# has 0 prime(s)

primorial  -------------> 67# has 0 prime(s)

primorial  -------------> 71# has 0 prime(s)

 Q2. What is the largest primorial you can find producing not one prime? primorial  -------------> 9133# has 0 prime(s)
 Q3. What primorial can you find producing the largest number of primes? primorial  -------------> 3# has 1 prime(s) primorial  -------------> 5# has 2 prime(s) primorial  -------------> 7# has 3 prime(s) primorial  -------------> 13# has 4 prime(s)

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Farideh Firoozbakht wrote:

Answer to Q1 : No, for the primorials prime(1)*prime(2)*...*prime(35) ,
prime(1)*prime(2)*...*prime(40) and ... there is no at least a prime.

Answer to Q2 : WE can find many large primorials with more than one primes.

Answer to Q3 : From each of the primorials prime(1)*prime(2)*...*prime(138)
and  prime(1)*prime(2)*...*prime(252) we get 8 primes.

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JK Andersen wrote:

The solution part says "BTW, someone has given incorrect answers".

That was also my first thought but then I realized Torbjörn Alm and Farid Lian
must have made another interpretation of the puzzle.
It says "replace succesively each '*' with a '+', one at the time", and then
gives an example where this procedure successively gives primes. The example
stops after the last '*' has been replaced but this happens to be at the first
fail. Torbjörn and Farid apparently thought that the replacement procedure
always stops at the first fail.

I did not submit a solution to this puzzle but I interpreted it as the
replacement procedure always continuing to the last '*'.

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