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 p2^k
is prime for 2<=2^k<p 
Erdos, p.42, A19, R.K.
Guy, UPiNT 
11 
The only
known example where p divides (p  1)#
 1. 
Lhun
/ (PC!) 
13 
13 = 2^2 +3^2 is the only
prime which is expressible as the sum of the squares of
two successive primes. 
Ken Wilke 
17 
Is the only integer
which is equal to the number of prime partitions of itself. 
Rupinski / (PC!)

37, 2 
only primes p such that phi(p1)=pi(p) 
Faride Firoozbakht 
71 
is the only known prime of the form n^prime(n)*prime(n)^n1,
next prime (if there exist) has more than 23580 digits. 
Faride Firoozbakht 
73 & 3 
3 & 73 are the only known primes of the
form n^prime(n)*prime(n)^n+1, next prime (if there exist) has more than
35323 digits 
Faride Firoozbakht 
83, 11 & 2 
There are only
three
known
primes p1 that satisfies the congruence p2.p31 mod p1 =0 (p1, p2 & p3
are consecutive primes) 
C. Rivera 
97, 13, 5

There are only 3 known
primes of the form 2^(2^m) + 3^(2^m), for m=0, 1 & 2. 
C. Rivera & Ken Wilke.
See Table 12 of "Prime Numbers and computer
Methods of factorization" by Hans Riesel. P. 417. 
101, 11 
There are only two
known primes of the form:10^n + 1. When n=1
and n = 2. 
Luis Rodríguez 
137 
is the only known number m such that,
pi(1)*1^1 + pi(2)*2^2 + ...+ pi(m)*m^m is prime.

Faride Firoozbakht 
241 
is the only number m such that p(m) =
pi(m*pi(m)) 
Faride Firoozbakht 
257, 5, 1 
There are only 3 known
primes of the form n^n+1, for n=1, 2 & 4 [or (2^m)^(2^m)+1, for m=0, 1 &
2] 
C. Rivera / p. 82.
"Excursions in
Number Theory", C.
Stanley O. & J. T. Anderson, Dover. 
563, 13, 5 
Are the only
known Wilson primes are
such that (p1)!=1(mod p^2) 
p. 29, "Prime Numbers,
a computational perspective". R. Crandall & C. Pomerance. 
2003 
Is
the only prime of the form 2*10^p+p,where p and p+4 is prime 
Faride Firoozbakht 
3511 & 1093 
Are the only
known
primes p that satisfies the congruence (2^p2) mod p^2 =0.
(These primes are also named Wieferich primes) 
p.9, A3, R.K. Guy,
UPiNT 
27941 
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). 
Luis Rodríguez 
65537, 257, 17, 5, 3 
The only
known Fermat primes 
P. Fermat 
56598313, 487 & 3 
Are the only
known
primes p that satisfies the congruence (10^p10) mod p^2 =0 
p.9, A3, R.K. Guy,
UPiNT

1418575498573, 97 
There are 2
known intervals that fulfill the Hardy's
conjecture pi(x,x+y)<=pi(y) That is: No interval of length y can contain
more primes than the first interval 0y.
a)
Interval=16 pi(x,x+16)=pi(16) = 6. Example x = 97 ; pi(97,113) = pi(16)
= 6
b) Interval=36 ; pi(x,x+36) = pi(36) = 11. Example
x = 1418575498573 
Luis Rodríguez 
.
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!! + ... + (p1)!! = 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 + ... + previousprime(p) = p (mod nextprime(p)).
In December she added:
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
***
Advised, Farideh wrote:
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
***
