February 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Re: HTML in math-fun, seqfan posts
by Dan Hoey 27 Feb '03

27 Feb '03

> I will never again take up the subjects of race, religion, politics or > HTML in math-fun or seqfan. I will not again post, or answer replies, > on this subject. :-) Thank you. Dan

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Re: [math-fun] Prime Question: Whoops
by Dan Hoey 27 Feb '03

27 Feb '03

Michael Kleber <kleber(a)brandeis.edu> writes: > No no no, these probabilistic arguments are going the > wrong direction. I figured out one reason last night, see below. > > Let a prime be a good prime if a digit can be deleted leaving either=20= > > > the empty string or a good prime. > > Are there infinitely many good primes? > > Let's be very naive and say that all we know is the prime > number theorem, Pr(n is prime) =~ 1/ln(n). > > Then E(# n-digit good primes) / E(# (n-1)-digit good primes) =~ > (# ways to add a digit) * (Pr(an n-digit number is prime), which > is around (10n) * (1/(n ln 10)) = 10/ln 10 =~ 4.34. So the number > of good primes should grow exponentially. That assumes that various n-1-digit primes you get have independent likelihood of being deletable. But that's not at all true, since if you have one n-1-digit delprime, then there is at least one n-2-digit delprime, and so the n-1-digit number you get from deleting the second digit first can be tranformed into the n-2-digit delprime by deleting a single digit. So the probabilities of getting delprimes by deleting different digits have a positive correlation. Looking at the sequence of deletions, I see that the case of divisibility by 3 is much different from what I had been thinking. The digits {1,4,7} and {2,5,8} must be deleted in alternation (possibly with intervening {0,3,6,9}s) or you will hit a number divisible by 3. In particular, the number of 1mod3 digits must differ from the number of 2mod3 digits by exactly 1. Dan Hoey(a)AIC.NRL.Navy.Mil

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Re: [math-fun] Re: HTML in math-fun, seqfan posts
by Henry Baker 27 Feb '03

27 Feb '03

David: There are a number of free email clients -- e.g., Eudora Lite (which I am using now) -- that can be downloaded and installed in a matter of minutes. If you only ever _send_ mail, but don't receive it, then it is trivial to set up -- you only need to put in an SMTP server. You don't even have to use the one you use with Outlook -- there are some public ones available. In fact, you can dispense with the SMTP server entirely for about $20 by using Mail Express Lite, which turns your _own_ machine into an SMTP server. (This message sent via Mail Express.) Regarding Outlook: you can save yourself an enormous amount of pain and suffering by avoiding Outlook, because most of the viruses/worms today target your Outlook file and forward the virus to everyone in your address book. No Outlook, no problem. Re MS Outlook taking over the world -- this is actually not true. More and more govt agencies, schools, etc., are throwing out MS in favor of Linux, OpenOffice, etc. So your arguments are not persuasive. If you use any kind of wireless device, the cost and hassle of dealing with anything but text messages becomes prohibitive. I would vote strongly for keeping ASCII text and barring html/rtf messages. MIME still works fine with text, and attachments still work fine. At 10:54 AM 2/27/03 -0500, David Wilson wrote: >I understand the tribulations of picking through MIME and HTML in a >nonsupportive mail reader. But., my current version of MS Outlook >includes HTML and MIME in outgoing messages by default, these >features are preferable for most of my mail; I have to explicitly switch >to plaintext mode for math-fun and seqfan postings, and sometimes >I forget. I can't select HTML or plaintext per recipient, consequently >slips will happen, and I can't be eternally apologetic. If people can't >overlook the occasional slip on this account, I can easily go away. > >At some point, I think math-fun and seqfan will have to acknowledge >that technology is moving on. Despite my personal distaste for >Microsoft and its applications, the reality is that MS Outlook is now >the most commonly-used email application, that it supports HTML >and MIME, and that users rely heavily on the associated benefits, such >as formatted and colored text, images, backgrounds, links, attachments, >etc. I appreciate that math-fun and seqfan like to be accomodating, >however, accomodating HTML-disabled users will increasingly disaffect >a growing majority of HTML-enabled users, and will deprive math-fun >and seqfan of some very useful technologies (when MathML browsers >become common, will you not take advantage?). > >I understand wanting to accomodate HTML-disabled users, and the >desire to save on server space; and undercurrents of nostalgia, >UNIXphilia, and MSphobia all warm my heart. It may also be that >math-fun and seqfan want to discourage a floodgate of membership, >considering some recent incidents of inconsiderate new users >disrupting our coffeehouse atmosphere. Notwithstanding, my >considered opinion is that catering to the technologically lowest >common denominator will become increasingly costly to math-fun >and seqfan as technology advances.

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Re: [math-fun] googol = ?
by Dan Hoey 27 Feb '03

27 Feb '03

David Wilson asked: > What would be the proper name for a googol? The biggest zillion I > know is 10^63 = 1 vigintillion = 1 20-zillion, this means that a > googol is 10 32-zillion, but I don't know what a 32-zillion is > properly called, if it has a name. The "Book of Numbers" naming system gives this as "ten duotrigintillion" US or "ten thousand sedecillion" British. I don't know why the answer from "Dr Math" is different. Dan Hoey

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[math-fun] Question
by David Wilson 27 Feb '03

27 Feb '03

Compare A040104 and A048051

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[math-fun] googol = ?
by David Wilson 27 Feb '03

27 Feb '03

What would be the proper name for a googol? The biggest zillion I know is 10^63 = 1 vigintillion = 1 20-zillion, this means that a googol is 10 32-zillion, but I don't know what a 32-zillion is properly called, if it has a name.

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[math-fun] Prime question
by David Wilson 26 Feb '03

26 Feb '03

Let p be a good prime if it has only one digit or else a single digit can be deleted leaving a prime. Should we expect there to be a largest good prime?

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Re: [math-fun] Prime question
by Dan Hoey 26 Feb '03

26 Feb '03

Don Reble <djr(a)nk.ca> writes: > > Let p be a good prime if it has only one digit or else a single > > digit can be deleted leaving a prime. Note that this definition was later amended. More below. > Sounds good to me. Indeed, a big prime may have a number of "shortened > primes"; I wonder how many to expect. > Roughly, a number N has a 1/log(N) chance of being prime. That is, > if a number has D digits, the odds are about log(10)/D = 2.302585/D. > If a number is prime, it's coprime to 10, at least, and so ends with > 1, 3, 7, or 9. And "almost all" of the shortenings are also coprime > to 10. The odds of being prime increase by a factor of almost 2.5. So that's the product of 1 = probability of staying prime to 2 2 = relative density of primes in {1} mod 2 1 = probability of staying prime to 5 5/4 = relative density of primes in {1,2,3,4} mod 5 so 5/2 log(10)/D = 5.756463/D. In general, consider that we delete the k'th digit, b, from a D-digit prime P=a + b*10^k + (c*10^(k+1)), producing N = a + c*10^k = P - (b + 9*c)*10^k. For any a prime q, N==0 (mod q) iff P==(b+9c)10^k (mod q). I get Prob(N==0) (mod q) = 0 when q=2 or q=5, 3/10 when q=3, 1/q when q>5. So the probability that N is prime will be (Log(10)/D) Product (1 - Prob(q|N))/(1-1/q) (q prime) = (Log(10)/D) * (1/(1/2)) * ((7/10)/(2/3)) * (1/(4/5)) * ((1-1/q)/(1-1/q)) ^ whatever = (21/8) (Log(10)/D) = 6.044286/D . > Anyway, although the odds might not scale by 2.5, it comes awfully > close: maybe 2.4971498... :-) So the primality-probability of a > shortened prime is about 5.7499001/D. > Now, a (D+1)-digit prime has D+1 shortenings to D-digit > numbers. You'd expect about 5.7499001*(D+1)/D primes among them. For > large D, that approaches just 5.7499001. > One might imagine the distribution of prime shortenings to be Poisson. > Therefore the probability that a big prime is _bad_ comes to > exp(-5.7499001). I think this should be 10^(-21/8) = .00237137. But the important thing is it's a constant. This is the probability that there is no way to remove a single digit and get a prime. But the intended definition was the probability we could remove a single digit and get a _good_ prime. If the probability of being able to remove a digit a second time were independent of the number of ways of removing the first digit, this would give a probability of .99762863^D that a prime would be good, and this would approach zero. In fact, we would expect a finite number of good primes. But to me it seems more likely that B(D), the probability of a D-digit prime being bad, might be (B(D-1))^((21/8) (Log(10)) = B(c) ^ (((21/8) Log(10))^(D-c)) which would make almost all large primes good. Dan

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Re: [math-fun] Prime question
by Mike Stay 26 Feb '03

26 Feb '03

Was that calculation done with this definition, or the corrected one he posted afterwards? (I.e. deletion of a digit yields another *good* prime, not just a prime.) >On Tue, 25 Feb 2003, David Wilson wrote: > >> Let p be a good prime if it has only one digit or else a single digit can be deleted leaving a prime. >> Should we expect there to be a largest good prime? >> > >My guess is no. I chose 10 random 200 digit primes and found that >5 of them were good primes. I did it again and found that out of >10 all 10 were good! > >This was done using Maple's probabilistic primality test "isprime". >Actually in each case I first selected a ramdom 200 digit positive >integer and then used the next 10 primes in a row to test for >goodness. > >--Edwin > > > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > -- Mike Stay staym(a)clear.net.nz

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Re: [math-fun] Lighten up, guys
by BobMayans＠aol.com 25 Feb '03

25 Feb '03

Well, I thought the song was funny, and made points worth making. I'm sure everyone's politics can withstand an occasional Bronx cheer. (I hope so, anyway). Bob M.

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