20070801, 01:14  #1 
∂^{2}ω=0
Sep 2002
República de California
5×17×137 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. 
20070801, 18:11  #2 
May 2004
New York City
5×7×11^{2} Posts 
Lemma: The probability that a random prime p is a number n is
equal to the probabilty that p1 is a number n1. Proof: Obvious. Having thus reduced the problem to a much simpler one, and allowing for infinite regress, the original problem is solved. 
20070801, 18:26  #3 
∂^{2}ω=0
Sep 2002
República de California
26575_{8} Posts 
Your proof is sound, but form an aesthetic viewpoint, I've never really liked proofs by induction. There's just something too brutishforce about them for my taste. But ... we all have our quirks.
Also, even glossing over the ambiguous nature of proofbyobviousness 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 20070801 at 18:30 
20070801, 18:43  #4 
Jun 2003
The Texas Hill Country
3^{2}×11^{2} Posts 
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 20070801 at 18:45 Reason: ewmayer "beat me to it". We are basically saying the same thing. 
20070801, 18:50  #5 
"William"
May 2003
New Haven
2·7·13^{2} 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 20070801 at 18:53 Reason: Added Google stats 
20070801, 18:58  #6 
May 2004
New York City
5×7×11^{2} Posts 
I would think that with the great computational skills evident
on this forum that the following derivation would be considered excessive: Let P_{p} be the probability that a random prime p is a number n. By the lemma, P_{p} = P_{p1} = ... . Hence multiplying the P_{p} gives (P_{p})^{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 20070801 at 19:05 Reason: details, details 
20070801, 18:59  #7  
∂^{2}ω=0
Sep 2002
República de California
5×17×137 Posts 
Quote:
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 noninductive, it is also noncapacitative and nonresistive. A sort of roomtemperaturesuperconducting proof, one might [humbly] say. 

20070801, 19:08  #8  
∂^{2}ω=0
Sep 2002
República de California
2D7D_{16} Posts 
Quote:
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20070801, 19:20  #9 
May 2004
New York City
5×7×11^{2} 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 20070801 at 19:24 Reason: additional detail 
20070801, 19:26  #10  
Jun 2003
The Texas Hill Country
3^{2}·11^{2} Posts 
Quote:
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. 

20070801, 20:11  #11  
Nov 2004
2^{2}·3^{3}·5 Posts 
Quote:
Norm Last fiddled with by Spherical Cow on 20070801 at 20:12 

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