**Faride Firoozbakht** has defined
three similar functions defining classes of integer numbers g(n,m) that
conjecturally containing at least one prime for each n>1 in the proper and
defined range for each function.

Here, I will only present one of these
functions - fi(n, m) in her original nomenclature -, the one that I consider more elegant by simple and general
^{(1)}:

- g
**(n,
m) = (n).m.(n-1).m.(n-2).m.
... (2).m.(1)**,
n>1, m=>0
- In this definition the dots "."
means concatenation of the numbers aside.
**Conjecture**:
For each n>1, there exists at least one positive m, 0<=m<n^2, such that
g(n,m) is prime.

As a matter of fact, **Faride** has
already computed the earliest m values that validate her conjecture for each
n<300.

In particular for each 2<=n<=20 here
are these m values: 1, 5, 14, 5, 5, 9, 1, 1, 29, 23, 28, 13, 46, 22, 18,
116, 35, 18, 155,... (See
A083660)

This means, for example, that **
7969594939291
**is the earliest prime for g(7,m).

**Question:**

**1)**** Can you
argument why this conjecture works, or to find a counterexample to it?**

**2) Do you think that
this kind of functions are interesting?**

_____

^{(1)
}See this and the other
two functions in the sequences
A083660
A082469 and
A083677

**Luis Rodríguez** wrote:

1.- The conjecture has a high probability of being
true because the mean value of its frequency is almost = n.

Demonstration:

Let be d = Number of digits of g(n,m)

Log = Neperian
logarithm

log = Decimal logarithm

dm = Aprox. number of digits of m.

dm = log(m) +1

N = g(n,m)

d= n + (n-1)dm = n(1+dm)-dm

Log(N) = Log(10^d) = 2.3d aprox.

Log(N) = 2.3n(1+dm) - 2.3dm

But 1/Log(N)= Probability of N being prime

As m can go from 1 to n^2 the mean value of
frequency of N being prime is = n^2/Log(N). The numbers N always end in 1,
so its probability is 2.5 times greater. Mean value = 2.5n/(2.3(1+dm))
aprox.

Example:
If n = 300 , Mean Value = 300/3.48 aprox. Expected
number of primes = 86

2.- This a simple puzzle without arithmetical
interest or consequences.

***