## FactoProths and PrimoProths

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.

and some curiosities (not proths) :

 Type Digits Who ? 6792!*2^4838+1 24536 Zoe Brown-Harvey 2613!*21004+1 8099 Didier Boivin 239#*2239+1   239#*2239-1 Twin 168 20#*220+1   20#*220-1 Twin 14 7#*27+1   7#*27-1 Twin 5 Henri Lifchitz 37#*231+1  37#*232+1   37#*233+1 Cunn.L=3 23 41#*29+1  41#*210+1   41#*211+1  41#*212+1 Cunn.L=4 18 Henri Lifchitz 32#*232+1  et 32!*232-1 21,46

You can also consult the next links :

Created by  Henri Lifchitz : February, 28 1999, last modification: October, 29 2002.