## Prime Formulas

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• Hello all: I ve gotten all these formulas are verified for ALL Prime Numbers based on the harmonic numbers. Pn = Floor [H [(1 +1 / Pn) ^ ((Pn) ^ 2)]-EulerGamma
Message 1 of 1 , Oct 29, 2011
Hello all:

I've gotten all these formulas are verified for ALL Prime Numbers based on the harmonic numbers.

Pn = Floor [H [(1 +1 / Pn) ^ ((Pn)
^ 2)]-EulerGamma +3/2 - n * (log n) / (Pn)
-1/n]

For all n> = 1, ie equality is true for all
primes.

Other formulas that
I have obtained are as follows:

Pn =Floor [H [(1 +1 / Pn) ^ ((Pn)
^ 2) / E ^ ((log n) +EulerGamma-(Pn)
/n -1/2 +3/(2n))]

And this one:

Pn = Floor [H [(1 +1 / Pn) ^ ((Pn)
^ 2) / E ^ (H
[n] - (Pn) / n-1 / 2 +1 / n)]

The three are verified for all prime numbers.

All of them are based on the following result which I obtained:

H [(1 +1 / n) ^ (n ^ 2)] = n + EulerGamma - 1/2 +Epsilon  with Epsilon -> 0 for all sufficiently large positive integer n. But ndoes not
need tobevery large so that the convergence is
good.

Floor [x] = The higest integer equal
or less than x.

EulerGamma = 0.57721566 .... Euler Mascheroni constant.

E = 2.71828182845 ... ...

Pn = nth prime number.

Log [n] = Ln [n]

H[x]= HarmonicNumber[x]

MATHEMATICA FILE:

F[n_]:=Floor[HarmonicNumber[(1+1/(Prime[n]))^(Prime[n]^2)/E^(HarmonicNumber[n]-Prime[n]/n-1/2+1/n)]]
G[n_]:=Floor[HarmonicNumber[(1+1/(Prime[n]))^(Prime[n]^2)/E^(Log[n]+EulerGamma-Prime[n]/n-1/2+3/(2n))]]

FG[n_]:=Floor[HarmonicNumber[(1+1/Prime[n])^((Prime[n])^2)]-EulerGamma+3/2-n
Log[n]/Prime[n]-1/n]
Do[Print[FG[n]," ",F[n],"
",G[n]," ",Prime[n]],{n,1,50,1}]
2   2   2   2
3   3   3   3
5   5   5   5
7   7   7   7
11   11   11   11
13   13   13   13
17   17   17   17
19   19   19   19
23   23   23   23
29   29   29   29
31   31   31   31
37   37   37   37
41   41   41   41
43   43   43   43
47   47   47   47
53   53   53   53
59   59   59   59
61   61   61   61
67   67   67   67
71   71   71   71
73   73   73   73
79   79   79   79
83   83   83   83
89   89   89   89
97   97   97   97
101   101   101   101
103   103   103   103
107   107   107   107
109   109   109   109
113   113   113   113
127   127   127   127
131   131   131   131
137   137   137   137
139   139   139   139
149   149   149   149
151   151   151   151
157   157   157   157
163   163   163   163
167   167   167   167
173   173   173   173
179   179   179   179
181   181   181   181
191   191   191   191
193   193   193   193
197   197   197   197
199   199   199   199
211   211   211   211
223   223   223   223
227   227   227   227
229   229   229   229

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