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 2007-08-01, 01:14 #1 ewmayer ∂2ω=0     Sep 2002 República de California 22×5×587 Posts odds of a random prime being a number I think the complement to this thread deserves its own thread. I have a marvelous proof of this, but am waiting for AMS to acknowledge receipt of manuscript before making it public.
 2007-08-01, 18:11 #2 davar55     May 2004 New York City 5·7·112 Posts Lemma: The probability that a random prime p is a number n is equal to the probabilty that p-1 is a number n-1. Proof: Obvious. Having thus reduced the problem to a much simpler one, and allowing for infinite regress, the original problem is solved.
 2007-08-01, 18:26 #3 ewmayer ∂2ω=0     Sep 2002 República de California 267348 Posts Your proof is sound, but form an aesthetic viewpoint, I've never really liked proofs by induction. There's just something too brutish-force about them for my taste. But ... we all have our quirks. Also, even glossing over the ambiguous nature of proof-by-obviousness and assuming what you say is true, your lemma only shows that the probability is the *same*, not what the probability *is*. Perhaps a corollary or a separate claim/lemma/theorem is in order. Last fiddled with by ewmayer on 2007-08-01 at 18:30
2007-08-01, 18:43   #4
Wacky

Jun 2003
The Texas Hill Country

32×112 Posts

Quote:
 Originally Posted by davar55 Having thus reduced the problem to a much simpler one, and allowing for infinite regress, the original problem is solved.
I don't think so. For an inductive proof, such as you suggest, you need two elements. You need the inductive step, such as you have indicated, but you also need a boundary (terminal) condition. You have failed to provide this portion of your "proof".

Last fiddled with by Wacky on 2007-08-01 at 18:45 Reason: ewmayer "beat me to it". We are basically saying the same thing.

 2007-08-01, 18:50 #5 wblipp     "William" May 2003 New Haven 45048 Posts A random Prime might turn out to be the Prime of Miss Jean Brodie or a Prime Rib Steak. I'm pretty sure neither of these is a number, so the probability in question appears to be less than 1. Googling "Prime" gets 225 million hits, "+Prime +Number" gets 142 million hits, so my guess is that the probability of a random prime being a number is 63%. William Last fiddled with by wblipp on 2007-08-01 at 18:53 Reason: Added Google stats
 2007-08-01, 18:58 #6 davar55     May 2004 New York City 10000100010112 Posts I would think that with the great computational skills evident on this forum that the following derivation would be considered excessive: Let Pp be the probability that a random prime p is a number n. By the lemma, Pp = Pp-1 = ... . Hence multiplying the Pp gives (Pp)n --> 1 or 0 depending on whether there exist any primes. Additional Lemma: There are primes! Proof: Start counting at 1 and continue until a number is reached whose only factors are (well, you know). This process terminates at p=2. Hence there are primes! Corollary: The desired probability is 1 (if there really are primes). Last fiddled with by davar55 on 2007-08-01 at 19:05 Reason: details, details
2007-08-01, 18:59   #7
ewmayer
2ω=0

Sep 2002
República de California

101101110111002 Posts

Quote:
 Originally Posted by Wacky I don't think so. For an inductive proof, such as you suggest, you need two elements. You need the inductive step, such as you have indicated, but you also need a boundary (terminal) condition. You have failed to provide this portion of your "proof".
Aha - a hole in davar55's "proof"!

See, I told you it was not so simple after all - which is why I am carefully refraining from revealing any of the power, the glory, the subtle elegance [the proofistic Feng Shui, if you will] that is my proof until I am sure it has been received and begun the peer review process. [As in, the referee says, "let me peer at it and get back to you..."]

The truly marvelous thing about my proof is that not only it is non-inductive, it is also non-capacitative and non-resistive. A sort of room-temperature-superconducting proof, one might [humbly] say.

2007-08-01, 19:08   #8
ewmayer
2ω=0

Sep 2002
República de California

22×5×587 Posts

Quote:
 Originally Posted by davar55 Let Pp be the probability that a random prime p is a number n. By the lemma, Pp = Pp-1 = ... .
Ah, but that assumes that for any number, subtracting 1 also gives a number. That is quite plausible, but also requires proof, to avoid the "it's plausible, so must be true" logical-fallacy trap.

Quote:
 Hence multiplying the P[sub]p gives (Pp)n --> 1 or 0
This assumes the sequence terminates - can you prove that it in fact always does? [And that every intermediate term is a number?]

Quote:
 Additional Lemma: There are primes! Proof: Start counting at 1 and continue until a number is reached whose only factors are (well, you know). This process terminates at p=2. Hence there are primes!
No, you have basically just "proved" that "2 is prime, because its only prime factor is 2, which is prime." In other words, a tautology, not a proof. I'm afraid that it's back to the drawing board with you my friend, despite your valiant and praiseworthy effort.

 2007-08-01, 19:20 #9 davar55     May 2004 New York City 5·7·112 Posts Well, thus begins (and perhaps ends) the review process. The necessary intermediate steps to complete my proof might take volumes, and perhaps a lifetime to solve a problem that has already been solved by another (albeit the solution is not yet revealed -- we await patiently). (Must prove 2 is prime ... must prove 2 is prime ... must prove 2 is prime ... ... Does this EVER terminate?) Last fiddled with by davar55 on 2007-08-01 at 19:24 Reason: additional detail
2007-08-01, 19:26   #10
Wacky

Jun 2003
The Texas Hill Country

32·112 Posts

Quote:
 Originally Posted by wblipp A random Prime might turn out to be the Prime of Miss Jean Brodie or a Prime Rib Steak. I'm pretty sure neither of these is a number
William,

I am in complete agreement with your conclusion that the probability is less than unity.

However, beware, I do not agree with your above statement. I have known a few "Misses" who certainly were "numbers", and d*mn good looking ones at that.

2007-08-01, 20:11   #11
Spherical Cow

Nov 2004

54010 Posts

Quote:
 Originally Posted by ewmayer Aha - a hole in davar55's "proof"! See, I told you it was not so simple after all - which is why I am carefully refraining from revealing any of the power, the glory, the subtle elegance [the proofistic Feng Shui, if you will] that is my proof until I am sure it has been received and begun the peer review process. [As in, the referee says, "let me peer at it and get back to you..."]
Ah- the ambiguity of the English language strikes again. I think the following is the proper "Pier Review" for the paper in question.

Norm
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Last fiddled with by Spherical Cow on 2007-08-01 at 20:12

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