Conjecture 103.  (6*prime(n)-1)/n

Sebastian Martin Ruiz sent the following conjecture, on Feb 19, 2023

Sea F(n)=(6 Prime(n) -1)/n

Conjetura: Si F(n) es entero entonces F(n) es primo

Estos son los valores enteros encontrados:


n F(n)
1 11 Prime

5 13 Prime

209 37 Prime

373 41 Prime

515 43 Prime

18815 67 Prime

117047 79 Prime

3526873 101 Prime

Q1) Probar que es cierta la conjetura o hallar un contraejemplo.
Q2) Hallar más valores enteros de F(n).

During the week 9-15 Dec 2023, contributions came from Giorgo Kalogeropoulos, Vicente Felipe Izquierdo, G. Resta, Alessandro Casini, Emmanuel Vantieghem, JM Rebert

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Giorgio wrote:

The conjecture is False.

First counterexample is:

(6*Prime(545855537) - 1)/545855537 = 133 = 7*19
 

For n=545855537, F(n)=133 which is NOT prime.

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Vicente wrote:

We can define the following simple equation:

n k + 1 = 6 p1

where "p1" = Prime[n] and "k" = F[n].

We can reach the following conclusion:

"n" is of the form 1+6x and "k" is of the form 5+6y or

"n" is of the form  5+6x  and "k" is of the form  1+6y

This greatly simplifies the search. All of Sebastian's solutions are one of those 2 ways.

Q1) The conjecture is False. The first counterexample is for: n = 5 + 6x90975922 = 545855537 where F[n] = 133, is not prime.

Q2) The sequence continues with n = 545855537

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Giovanni wrote:

Counterexample:

prime(545855537) = 12099797737 and  (6*12099797737 - 1) / 545855537 =
133, which is not a prime number.

the next integer values are for prime(1416486707) = 32815275379 which
gives F(n)=139 which is prime and
for prime(6971972953)= 173137328333 which gives F(n)=149 which is also
prime.

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Alessandro wrote:

The conjecture is false. In fact,  the next number such that 6*prime(n)-1 is divisible by n is 545855537, which has a composite quotient (133). The following one instead satisfies the conjecture's property, that is 1416486707.

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Emmanuel wrote:

The conjecture is not true !

The next  n  for which  F(n)  is an integer is  545855537.

The value of  F  at  n  is  133 = 7*19, not prime.

I found two more values for which  F(n)  is an integer :

1416486707  and  6971972953.
 

The corresponding values of  F(n)  are :

139  and  149, both prime.

 

I found no more integer values for  F(n)  for  n <= 3*10*10.

 

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Rebert wrote:

Q1)

 F(545855537) = 133 = 7*19 is not prime

Q2) I found:

n F(n)

1 11 is prime

5 13 is prime

209 37 is prime

373 41 is prime

515 43 is prime

18815 67 is prime

117047 79 is prime

3526873 101 is prime

545855537 133 = 7*19 is not prime

1416486707 139 is prime

6971972953 149 is prime

 

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