The developed idea here is to combine properties of Proth numbers k*2^{n }+/ 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 :
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 BrownHarvey  
2613!*2^{1004}+1  8099  Didier Boivin  
239#*2^{239}+1 239#*2^{239}1 
Twin 
168


20#*2^{20}+1 20#*2^{20}1 
Twin 
14


7#*2^{7}+1 7#*2^{7}1  Twin  5  Henri Lifchitz 
37#*2^{31}+1 37#*2^{32}+1 
Cunn.L=3 
23


41#*2^{9}+1 41#*2^{10}+1 41#*2^{11}+1 41#*2^{12}+1 
Cunn.L=4  18  Henri Lifchitz 
32#*2^{32}+1 et 32!*2^{32}1 
21,46

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