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 A083660)

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


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.


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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