Problems & Puzzles: Puzzles

Puzzle 225. VSP (Very Sparse Primes)

Here we will ask for these properties satisfied by only one prime (preferred) or a very small set of primes (let's say, less than 6 members).

We will discard "digital properties", that is to say, properties that may disappear if the prime is expressed in a base other than 10.

We may report true or absolute "only" primes (if it has been demonstrated that no other  primes share this property).

We may also report "only known" primes (if that demonstration does not exist at the date of the report).

Evidently that all these small sets of "only known" primes, for a given property, are an invitation to discover the next one or the demonstration of the inexistence of more solutions.

I will start providing only a few examples of these properties just to give an idea of what kind of properties are wanted here.

The reader is invited to submit other properties that he consider interesting.

 Prime(s) Property Author and/or Reference 2 Is the only prime such that the sum of its divisors is another prime ? 3 Is the only number that is prime and triangular Is the only known number where n^n - n + 1 forms a square number The only prime, p, such that p + 1 is a square. Gupta / (PC!) Lhun / (PC!) ? 5 The only prime which is the difference of two prime squares. Gupta / (PC!) 7 Is the only known prime p such that every number p-2^k is prime for 2<=2^k

.

Solution: The very first contributions came from Luis Rodríguez and Ken Wilke (July 2003). These can be found in the Table above. Hopefully more will be reaching as the time goes on.

***

Martin Boucher sent the following message on Dec. 2004:

Asunto: Prime Puzzle 225 - Very Sparse Primes

Mensaje: To add to the :
'There are two known values of A for the trinomial X^2 + X + A ,that
produces 600 or more primes when x runs from x = 0 to 1000. They are :
A = 247757 (composite) and A=27941 (prime). Note: Faride Firoozbakht
claims that A=595937  (composite) also provides more than 600 primes (609).'

I agree with Faride's A=595937:

I also have the following:

A=115721 - 627
A=55661 - 620
A=72491 - 611
A=136517 - 611
A=41537 - 606

Nothing seems to come close to the 657 that you get with the already
mentioned A=247757. Apologies if this is repeating what someone else has already sent- I could only find the ref above to 'two known values'

***

Several more contributions came from Farideh Firoozbakht on Nov 09:

2 is the only prime of the form n^n - 25.

2 is the only known prime of the form n^n - (2m - 1)^2.

3 is the only known number n such that n^n - 2 is square.

5 & 7 are the only known primes p (up to 10^5) such that
1! + 2! + ... + (p - 2)! = p - 1 (mod p).

3 & 11 are the only known primes p (up to 8*10^5) such that
1!! + 2!! + ... + (p-1)!! = p (mod p+1).

7 is the only known prime of the form n^n + 6 (next such
prime, if it exists has more than 17000 digits).

19 is the only known prime of the form n^n - 8, (next such
prime, if it exists has more than 25000 digits)

3119 is the only known prime of the form n^n - 6 (next such
prime, if it exists has more than 17000 digits).

16777213 is the only known prime of the form n^n - 3 (next such
prime, if it exists has more than 17000 digits).

5, 37 & 6569 are the only known primes p (up to 10^10) such that
2 + 3 + 5 + ... + previous-prime(p) = p (mod next-prime(p)).

We can easily show that :

1. 97 is the only prime of the form n!! - pi(n)!! .

2. 2, 3 , 5 & 113 are the only primes of the form n!! + pi(n)!! . So 113 is the
only mutidigit prime of the form n!! + pi(n)!! .

***

Giovanni resta wrote:

I noticed the last contribution to the puzzle by Farideh, in December.
She wrote:

1) 97 is the only prime of the form n!! - pi(n)!! .
2) 2, 3, 5 & 113 are the only primes of the form n!! + pi(n)!!.

Actually, it seems to me that, because,
9!! = 945 and pi(9)=4 and 4!! = 8, we have that

1) there is a second and larger prime of the form n!! - pi(n)!!, i.e.,
9!! - pi(9)!! = 937

2) there is a further larger prime of the form n!! + pi(n)!!, i.e.,
9!! + pi(9)!! = 953

***

Many thanks for telling me my mistakes and sorry for these mistakes.

The correct message is as follows.

We can easily show that :

1. 97 & 937 are the only primes of the form n!! - pi(n)!! .

2. 2, 3 , 5, 113 & 953 are the only primes of the form n!! + pi(n)!! .
So 113 and 953 are the only multidigit primes of the form n!! + pi(n)!! .

Proof : Numbers n such that n <= 12 and n!! + pi(n)!! is prime are 7 & 9 and the
corresponding primes are 97 & 937.
Also numbers n such that n <= 12 and n!! - pi(n)!! is prime are 1, 2, 3, 7 & 9 and
the corresponding primes are 2, 3 , 5, 113 & 953.
We show that for n > 12 both numbers n!! - pi(n)!! & n!! + pi(n)!! are composite.
It's obvious that for n > 5 , 3 divides n!! , so if pi(n) > 5 , 3 divide pi(n)!!
(3 is in the set of odd numbers less than pi(n), also 3 divides at least one of the
even numbers up to pi(n)).
Now, since for n > 12 , pi(n) > 5 hence for n >12, 3 divides both numbers n!! & pi(n)!! .
So for n > 12, 3 divides both numbers n!! - pi(n)!! & n!! + pi(n)!! , namely they are
composite.

***

On August 2013, Jahangeer Kholdi & Farideh Firoozbakht wrote:

19 is the only known solution of the equation sigma(x - 1) = 2x + 1.
There is no any other solution up to 2*10^9

*** Records   |  Conjectures  |  Problems  |  Puzzles  Home | Melancholia | Problems & Puzzles | References | News | Personal Page | Puzzlers | Search | Bulletin Board | Chat | Random Link Copyright © 1999-2012 primepuzzles.net. All rights reserved.