Problems & Puzzles: Conjectures

Conjecture 35. The Firoozbakht functions

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

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 A082469 and

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 = N
eperian 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.

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