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:

(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. ***