We denote whole numbers $ m> 1 $ share the product of different primes $ m $ as $$ operatorname {rad} (m) = prod_ { substack {p mid m \ p text {prime}}} p, $$

with the definition $ operatorname {rad} (1) = 1 $ (see you want the Wikipedia *Radical of an integer*), this is the famous multiplicative function in the statement of the abc conjecture. We also call that $ k $th primorial as $$ N_k = prod_ {t = 1} ^ k p_t $$ this is the product of the first $ k $ Primes.

I was inspired in the first paragraph of the section **B46** (1) to propose a variant of the type of problem the book presents. I think about solutions $ (p, k) $, from where $ p $ always denotes a prime number for which the identity

$$ frac {p-1} { operatorname {rad} (p-1)} = N_k tag {1} $$

applies to some primitive ones $ N_k $With $ k geq 1 $,

I've used a Pari / GP program to write the first satisfying primes $ (1) $, these first few primes $ p $are (this is a selection of my calculations) $ 5,13,29,37,53,61,149,157,173,181,229, 269,277, $ $ 293,317, ldots $ which correspond to these indices $ k $& # 39; s

$ 1,1,1,2,1,1,1,1,1,2,1,1,1,1 ldots $ our origins $ N_k $ fulfill the equation $ (1) $,

The curiosity that I have is something about the size or cardinality of the set $$ mathcal {K} = {k geq 1: (p, k) text {is a solution to the equation} (1) }. $$

**Guess.** *The sentence* $ mathcal {K} $ *is finally*,

So what I'm saying is the set $ mathcal {K} $ remains as a finite / finite set of positive integers, if the other variable $ p $ Runs over the set of all primes.

**Question.** Can one refute earlier assumptions? Can you do some thinking or calculations

clarify the truthfulness of the conjecture? **Many thanks.**

Maybe they can be useful state boundaries or an approximation to cardinality $ | mathcal {K} | $ for increasing segments of primes $ 2 leq p leq X $ and primors with indices $ 1 leq k leq Y $

Under these circumstances, I have no idea whether previous guesses are true, but I said I just want to find solutions to the integers $ k = $ 1.2 and $ 3 $ (I do not know if my contribution to this sequence has good mathematical content, the comments are welcome).

## references:

(1) Richard K. Guy, Unresolved Problems in Number Theory, Unresolved Problems in Intuitive Mathematics Volume I, 2nd Edition, Springer-Verlag (1994).