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[math-fun] Irrational-base representation
by asimovd＠aol.com 04 Mar '03

04 Mar '03

I recently began wondering if anything special happens when one uses the scheme of decimal, binary, etc. representation -- as usual, via a sequence of integer coefficients no greater than the base -- but with an irrational base like e. Of course, the representation has much more leeway for nonuniqueness than for an integer base, unless an algorithm is specified making it unique. So, we use the obvious "greedy" algorithm to get a well-defined representation. With base e, heuristically, a generic number's representation would in the long run have fraction 1/e of its digits = 0, 1/e of its digits = 1, and (e-2)/e of them = 2. I've taken a look at a few numbers base e but haven't found any statistical or other abnormalities. For example, I don't know what distinguishes an integer >= 2 from a noninteger, a rational from an irrational, a 1/prime from 1/nonprime, etc. Without evident pattern, it's still amusing to see pi expressed "base e". It starts: pi = 10.101002020002111120020101120001010200010012101202012100211200... Does anyone know of research on representing numbers via real irrational bases? --Dan

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[math-fun] permutations with cycles devoid of sequential elements
by Meeussen Wouter (bkarnd) 03 Mar '03

03 Mar '03

take P to be a permutation on the integers, with a cycle representation so that, in any cycle, if k is an element, neither k+1 nor k-1 are. How to count these non-neighbouring P's ? Wouter. =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] Prime Question: Whoops
by David Wilson 02 Mar '03

02 Mar '03

What I meant was, Let a prime be a good prime if a digit can be deleted leaving either the empty string or a good prime. Are there infinitely many good primes?

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[math-fun] (no subject)
by Richard Guy 28 Feb '03

28 Feb '03

Apologies for not having sent this earlier: Amongst our cards this season was one from Severo Ornstein, son of Leo Ornstein (1893--2002), well-known composer, and Laura Gould, daughter of Dick & Emma Lehmer, well-known mathematicians. (``Emma remains exactly the same at 96 -- strong and well, walking in the park, feeding her increasing horde of fat squirrels, ...'') Inside is a cartoon of Bush shouting `Yahoo' and riding a missile labelled `Iraq or bust!' and the following (Dr. Strangelove revisited): Sing this song to the tune of ``If You're Happy and You Know It Clap Your Hands'' If we cannot find Osama, bomb Iraq. If the markets hurt your Mama, bomb Iraq. If the terrorists are Saudi And the bank takes back your Audi And the TV shows are bawdy, Bomb Iraq. If the corporate scandals growin', bomb Iraq. And your ties to them are showin', bomb Iraq. If the smokin' gun ain't smokin' We don't care, and we're not jokin'. That Saddam will soon be croakin', Bomb Iraq. While the globe is slowly warming, bomb Iraq. Yay! the clouds of war are storming, bomb Iraq. If the ozone hole is growing, Some things we prefer not knowing, Though our ignorance is showing, Bomb Iraq. So here's one for dear old daddy, bomb Iraq. >From his favorite little laddyy, bomb Iraq. Saying no would look like treason. It's the Hussein hunting season. And it's oil that is the reason, Bomb Iraq.

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[math-fun] OOPS! Re: (Another) Prime Question
by asimovd＠aol.com 28 Feb '03

28 Feb '03

As stated, my question, except for p = 2, is nonsense, since multiplication by p and then adding one takes an odd number to an even one.. So -- what if we just pretend I never mentioned any prime but 2: << Consider the mapping f: N -> N given by f(n) = 2n + 1. Consider all infinite sequences (*) q_0, q_1, q_2, ..., where q_(n+1) = f(q_n) for n = 0,1,2,... 1. Is there an infinite prime sequence (*) ? 2. If not, are the lengths of such sequences unbounded? a) in any one sequence? b) if not, then perhaps among all possible such sequences? 3. If not, what's the length of the longest sequence of primes? a) in one sequence b) among all such sequences? 4. Is there even a sequence containing infinitely many primes? 5. If so, what's the least upper bound of the various densities of the primes in such sequences, and is it attained? >> --Dan A.

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[math-fun] (Another) Prime Question:
by asimovd＠aol.com 28 Feb '03

28 Feb '03

<< First, what about other bases besides ten? [This is normally dan asimov's question, isn't it?] >> I must express my gratitude for someone's finally taking the burden off me in this case (:-)>. In fact I was thinking of related questions that aren't directly base-related. Here's one: For any prime p consider the mapping f_p : N -> N given by f_p(n) = p*n + 1. Consider (possibly finite) sequences of primes (*) q_0, q_1, q_2, ..., where q_(n+1) = f_p(q_n) for n = 0,1,2,... Given prime p fixed: 1. Is there an infinite prime sequence (*) ? 2. If not, are the lengths of such sequences unbounded? 3. If not, what's the length of the longest sequence? And: How do the answers to the above questions depend on p ? (Variants: one could look at g_p(n) = p*n - 1, or allowing +-1 at each iteration, or even allow any of the f_p's (or g_p's) to be applied at any stage in the iteration.) But to start with, I'm most interested in the original question for p = 2. --Dan A.

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P.S. Re: [math-fun] (Another) Prime Question:
by asimovd＠aol.com 27 Feb '03

27 Feb '03

I wrote: << For any prime p consider the mapping f_p : N -> N given by f_p(n) = p*n + 1. Consider (possibly finite) sequences of primes (*) q_0, q_1, q_2, ..., where q_(n+1) = f_p(q_n) for n = 0,1,2,... >> Another question: If there are no infinite sequences (*) composed purely of primes, do there exist sequences (*) that *contain* infinitely many primes? If so, how large can the density of the primes be in (*) ??? --Dan A.

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[math-fun] A050444 correction
by David Wilson 27 Feb '03

27 Feb '03

Actually there are a few questions about A050444. First, there is the question of how spaces and dashes and the word "and" in number names affects the sequence. Certain features of the sequence tend to convince me that spaces and dashes and the word "and" are not included in the comparisons. This is consistent with "Knuth's Number" 8018018851 being the lexically first prime. But whatever the comparison criteria, there are definite errors in the sequence. For instance, the numbers 11 and 18049 are missing, as well as Knuth's number 8018018851. Anyway, here is what I get for A050444: 2 3 5 11 83 89 809 811 881 18013 18041 18049 18059 18089 18803 18839 18859 18000041 18000089 18000803 18000811 18000883 18000887 18000889 18018017 18018043 18018047 18018053 18018059 18018839 18018841 18018883 8000000011 8000000081 8000000809 8000000851 8000018009 8000018051 8000018087 8000018809 8000018881 8018000021 8018000059 8018000087 8018000839 8018000867 8018018009 8018018051 8018018081 8018018089 8018018819 8018018849 8018018851

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[math-fun] Re: HTML in math-fun, seqfan posts
by Christian G. Bower 27 Feb '03

27 Feb '03

I understand the importance of remaining on-topic in a list like this one and refraining from off-topic discussions of things people have strong opinions about (e.g. religion, politics, copyrights...) but are we at the point where the policy of the list itself is one of those taboo subjects? Members of a list ought to be able to question the list policy and make suggestions for change in a civilized way with the occasional (this is how we are doing it, discussion closed) type post from the moderator. Christian

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

27 Feb '03

I have submitted sequences for tile values and counts in Scrabble.

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