The developed idea here is to combine properties of Proth numbers k*2ⁿ +/- 1, with those of the factorial m!=2.3.4.5.6...m or the primorial m# =2.3.5.7.11...(p<=m) by introducing them in the expression of k :

1. Numbers m!*2ⁿ +/- 1 ("FactoProths") and m#*2ⁿ +/- 1 ("PrimoProths") have no small factors <= m.
2. For k*2ⁿ+1 and if k<2ⁿ we can use the déterministic primality test of Proth. In this case it will be necessary to link n to m! or to m#. We will also notice that the base for the test is > m.
3. The density of primes among these numbers is greater than for classic Proth numbers (k and n unattached), which allows new methods of research.

Choice of n in order to verify the condition of Proth's numbers k<2n :

For FactoProths :
By the utilization of the Stirling's formula for m! it comes approximately n > (m+1.5)*ln(m)/ln(2) - m*(1+1/ln(2)) + 2

For PrimoProths :
By the utilization of the Tchébycheff 's function  Théta(x)= approximately x, it comes approximately n > m/ln(2) - 1 

Generalization :
It is possible to replace the base 2 by an other base b, but the equivalent Proth  test becomes more complex.

Some first records

  and some curiosities (not proths) :

You can also consult the next links :

•  Prime Pages glossary factorial prime by Chris K. Caldwell.
•  Prime Pages glossary primorial by Chris K. Caldwell.
•  The top twenty: Primorial and Factorial Primes by Chris K. Caldwell.