- Hi All!

Consider the following sequence 0,0,0,1,1,2,2,3,5,5,7,8,8,9,11 ...

What are the next numbers?

Okay, okay you clever lot, yes it's easy to determine that the continuation is

13,13,15,16,16,18,19,21,24,25,25,26,26, 27,33,34,36,36,40,40,42 ...

Obviously you've all been doing too many MENSA tests.

Actually, and technically, the sequence comprises two alternating lead diagonals from Polya/Alisford infinite matrices, which can be easily reconciled and extended as far as you desire.

Welcome, then, to the Automatic Prime Number Finder, hereinafter known as APF. So how ( I hear you ask), can you possibly locate the primes directly from this APF sequence?

Well the mathematical reasons are deep, but the results, extremely practical and easy to apply. :-)

Let's inspect the first 30 numbers of the sequence

APF:0,0,0,1,1,2,2,3,5,5,7,8,8,9,11,13,13,15,16,16,18,19,21,24,25,25,26,26,27,33

Suppose you want to know what the 20th odd prime number is - well the 20th number in the above sequence is 16, and 20 + 16 = 36, which we'll call "T ". Now apply the function 2T+1, and voila, we have (2*36) + 1 = 73, the 20th odd prime number. Simple as that.

The following table demonstrates the stunning power and beauty of the arithmetic using this method:

APF No. T 2T+1

0 1 1 3 0 2 2 5 0 3 3 7 1 4 5 11 1 5 6 13 2 6 8 17 2 7 9 19 3 8 11 23 5 9 14 29 5 10 15 31 7 11 18 37 8 12 20 41 8 13 21 43 9 14 23 47 11 15 26 53 13 16 29 59 13 17 30 61 15 18 33 67 16 19 35 71 16 20 36 73 18 21 39 79 19 22 41 83 21 23 44 89 24 24 48 97 25 25 50 101 25 26 51 103 26 27 53 107 26 28 54 109 27 29 56 113 33 30 63 127 34 31 65 131 36 32 68 137 36 33 69 139 40

34 74 149 40 35 75 151 42 36 78 157

So now you don't need polynomial prime generating functions to find prime sequences, fancy formulae, Riemanns Hypothesis, nor even trial division - all you need is the APF index to find any and every prime number, except 2 of course.

Suppose you want to know what the 999,999,999th odd prime number is.

Simply get the APF index to generate the relevant sequence number; this can be done on PARI.PG, although PARI.Lo-Flo is faster, and as the latter name implies, stack overflows, when dealing with thousands of digits, are avoided.

PARI.Lo-Flo returns a result of 10,400,881,745 - add this to 999,999,999 to get T; perform the 2T+1 function and straightaway you"ll know that the 999,999,999th odd prime is 22,801,763,489.

It's my guess that this process has been known for years to both Bletchley Park and the NSA, and with it they have broken all the Internet Codes and RSA numbers at will. One thing's for sure I'm not going to use my credit card on the Internet any more.

Enjoy!

PS: If anyone can help co-write this remarkable finding, in order to publish a paper in formal mathematical jargon, so that even mathematicians can understand and grasp the concept, please let me know

[Non-text portions of this message have been removed] - --- Bob Gilson <bobgillson@...> wrote:
> Hi All!

Ahttp://www.research.att.com/~njas/sequences/A008507

>

> Consider the following sequence 0,0,0,1,1,2,2,3,5,5,7,8,8,9,11 ...

>

> What are the next numbers?

008507 Number of odd composite numbers less than n-th odd prime.

0, 0, 0, 1, 1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11,

> Welcome, then, to the Automatic Prime Number Finder, hereinafter known as

So you want to find primes using a sequence defined in terms of where the

> APF. So how ( I hear you ask), can you possibly locate the primes directly

> from this APF sequence?

primes are?

Phil

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http://mobile.yahoo.com/mail - --- In primenumbers@yahoogroups.com, Bob Gilson <bobgillson@...> wrote:

> It's my guess that this process has been known for years to both

Bletchley Park and the NSA, and with it they have broken all the

Internet Codes and RSA numbers at will. One thing's for sure I'm not

going to use my credit card on the Internet any more.

Putting aside the rest of your argument, a method to find all the

primes quickly would not cause breaking RSA (or any other encryption

system I'm aware of). Determining whether a specified number is a

probable prime is an easy problem (seconds for numbers of the size

used in RSA implementations), and a large probable prime is almost

certain by all conventional standards of probability to be prime. If

the NSA or anyone else found an efficient way to find the *prime

factors* of large numbers, then they'd be in business (at least in

terms of breaking RSA, which is being used somewhat less often for

other reasons). Posting to the list because this is a moderately

common misunderstanding.

Nathan