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)
***
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 nonzero size
***
Emmanuel Vantieghem wrote:
For most values of n (>2), at least one of the sums p_k#+(p_n#)/(p_k#)
(k = 1, 2, ...,n1) 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.
***
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.
***
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
***
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#)
This answers Q1 as well.
Q3: Changing the first k consecutive ‘*’s into ‘+’s:
No larger solutions than k=4, 13#.
The largest number of primeterms: 10, for 2129#.
Related: all splits are prime: only 3#, 5# and
7#.
***
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) 

***
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.
***
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 '*'.
***
