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?

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