Lambert Prime Number Formula
Copyright 2007, Carl L. Lambert
Note that a primorial is the product of all
prime numbers from 2 up to and including the
chosen prime number. Its math symbol is the
number sign (or pound sign, or hashtag, #). Also
the asterisk (*) is used herein for the
multiplication symbol. The example of 7 primorial
therefore is:--
7# = 2 * 3 * 5 * 7 = 210
The Lambert prime number formula is very
similar in appearance to the primorial formula
except herein the primorial itself is multiplied
by values other than 1 and wherein it is
incremented by values other than +1 or -1. It
has already been practiced to factor some titanic
numbers into such shorthand notation, but until
now only the Lambert formula states the ranges of
its three variables to produce all of the prime
numbers. The Lambert formula also produces some
composites but those composites are vitally
important to produce further primes, and all such
composites produced are systematically removed
during further progression of the formula's
iterations leaving a trail of nothing but the
primes, and ALL of the primes.
Here's the formula:
n * p# + v
(1) Where p is a prime number, with 2 being
arbitrarily considered the first prime.
The formula is used by assigning successive
values to p as found via its own
processing.
(2) Where n takes on all whole positive
integers from 1 to 1 less than the next p.
(3) Where v can always be 1 and is each of all
of the values produced by the formula
using all primes for p up to the
previous p ...
...and that are greater in value than
the current value of p.
...and that are not evenly divisible by
the current value of p.
...and whether or not they are prime.
...and all multiples of the current p
found at this point can now be
struck from the growing list of
results so far created. The first
such multiple will always be the
square of p and no multiples of the
current p will ever again appear in
any further iterations of the
formula. ONE MUST NOT YET REMOVE
ANY MULTIPLES OF ANY HIGHER PRIMES
FROM THE LIST.
To start:
p = 2
n = 1,2 (this is true even though we don't
yet know if 3 is prime)
v = 1
Therefore:
1 * 2# + 1 = 3
2 * 2# + 1 = 5
Next:
p = 3 (the next prime we've discovered)
n = 1,2,3,4 (note we've also discovered 5)
v = 1, 5
Therefore:
1 * 3# + 1 = 7
1 * 3# + 5 = 11
2 * 3# + 1 = 13
2 * 3# + 5 = 17
3 * 3# + 1 = 19
3 * 3# + 5 = 23
4 * 3# + 1 = 25 (be patient here)
4 * 3# + 5 = 29
Next:
p = 5
n = 1,2,3,4,5,6
v = 1,7,11,13,17,19,23,29 (here is where
all multiples of 5 are eliminated)
Therefore:
1 * 5# + 1 = 31
1 * 5# + 7 = 37
1 * 5# + 11 = 41
1 * 5# + 13 = 43
1 * 5# + 17 = 47
1 * 5# + 19 = 49 (be patient, 7 * 7)
1 * 5# + 23 = 53
1 * 5# + 29 = 59
2 * 5# + 1 = 61
2 * 5# + 7 = 67
2 * 5# + 11 = 71
2 * 5# + 13 = 73
2 * 5# + 17 = 77 (7 * 11)
2 * 5# + 19 = 79
2 * 5# + 23 = 83
2 * 5# + 29 = 89
3 * 5# + 1 = 91 (7 * 13)
3 * 5# + 7 = 97
3 * 5# + 11 = 101
3 * 5# + 13 = 103
3 * 5# + 17 = 107
3 * 5# + 19 = 109
3 * 5# + 23 = 113
3 * 5# + 29 = 119 (7 * 17)
4 * 5# + 1 = 121 (11 * 11)
4 * 5# + 7 = 127
4 * 5# + 11 = 131
4 * 5# + 13 = 133 (7 * 19)
4 * 5# + 17 = 137
4 * 5# + 19 = 139
4 * 5# + 23 = 143 (11 * 13)
4 * 5# + 29 = 149
5 * 5# + 1 = 151
5 * 5# + 7 = 157
5 * 5# + 11 = 161 (7 * 23)
5 * 5# + 13 = 163
5 * 5# + 17 = 167
5 * 5# + 19 = 169 (13 * 13)
5 * 5# + 23 = 173
5 * 5# + 29 = 179
6 * 5# + 1 = 181
6 * 5# + 7 = 187 (11 * 17)
6 * 5# + 11 = 191
6 * 5# + 13 = 193
6 * 5# + 17 = 197
6 * 5# + 19 = 199
6 * 5# + 23 = 203 (7 * 29)
6 * 5# + 29 = 209 (11 * 19)
Next:
p = 7
n = 1,2,3,4,5,6,7,8,9,10
v = 1,11,13,17,23,29,31,37,41,43,47,53,59,
61,67,71, 73,49,83,89, 97,101, 103,107,
109,113,121,127, 131,139, 143,151, 157,
163,167, 169,173, 179,181,187, 191,193,
197,199,209 (Note that 121, 143, 169,
187, and 209 are all composites that
MUST be used next, but all multiples
of the present value of p have been
eliminated, and they occur from p^2
upwards in consecutive prime multipliers
until the end of the formula's results
so far.)
Therefore:
1 * 7# + 1 = 211
1 * 7# + 11 = 221 (13 * 17)
1 * 7# + 13 = 223
...
1 * 7# + 121 = 331 (first prime created
that uses a composite for v)
... etc.
The elimination of multiples of the current
p being used reminds one of Eratosthenes' Sieve.
The real difference here is that, not only does
the user know exactly which composites to
eliminate, i.e., p * p, p * (next p), p * (2'nd
p after p), etc., but also those "few" multiples
of the current p are the ONLY ones he ever needs
to eliminate. Eratosthenes' Sieve needs to
eliminate ALL multiples of every p ALL the way
to infinity. It bears repeating that, once all
multiples of a current p are eliminated, no
multiples of that p will ever again occur in
further iterations of the formula.
It could also be noted that one can use the
Lambert formula to skip ahead of all initial
iterations to produce titanic primes. Merely
process the primorial of a large enough prime to
produce a multi-million digit number, multiply
it by any whole number from 1 up to 1 less than
the next larger prime than that used, and add
any nearby prime that is greater than the p
used. Of course one will not know what possible
composites may also be used for v in this case,
but at least many possible primes can be
forthcoming via such scheme. Caution! Any
result of the formula COULD BE a composite,
therefore it must be tested, and the testing of
titans is gruesome. The use of one of a set of
twin primes for the value of v in a calculation
of the formula, and the use of the other prime
of the set for v in a second calculation that
uses the same p and the same n will, of course,
possibly produce another twin set.
***
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