September 2006 - mathfun@mailman.xmission.com

[math-fun] First differences are primes
by Eric Angelini 25 Sep '06

25 Sep '06

Hello, could someone please extend this (if of interest for the OEIS) : S = 1 4 6 25 30 77 84 95 108 125 148 177 208 245 286 329 382 441 502 573 640 713 792 875 964 1065 1162 ... Definition : « Non-primes sequence whose first differences show all primes, once » The sequence S is build like this : Start with 1: S = 1 Add the smallest prime not added so far, in order to get a composite: - Can we add 2? No, because 1+2=3 and 3 is prime - Can we add 3? Yes, because 1+3=4 and 4 is composite So we have now: S = 1,4, Add the smallest prime not added so far in order to get a composite: - Can we add 2? (smallest available prime); yes, because 4+2=6 and 6 is composite So we have now: S = 1,4,6 - Can we add 5? No, because 6+5=11 and 11 is prime - Can we add 7? No, because 6+7=13 and 13 is prime - Can we add 11? No, because 6+11=17 and 17 is prime - Can we add 13? No, because 6+13=19 and 19 is prime - Can we add 17? No, because 6+17=23 and 23 is prime - Can we add 19? Yes, because 6+19=25 and 25 is composite So we have now: S = 1,4,6,25 - Can we add 5? (smallest available prime); yes, because 25+5=30 and 30 is composite S = 1,4,6,25,30 etc. Best, É. (I'm working on seq. T : « Primes sequence whose first differences show all non-primes, once » Hope this one is not in the OEIS either)

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Re: [math-fun] inconvenient directed graphs
by Daniel Asimov 25 Sep '06

25 Sep '06

Jim writes: << For each positive integer n > 0, what is the maximum value of N for which there exists an n-by-n stochastic matrix A and a probability vector v such that A^N v is the first forward image of v under repeated multiplication by A that lies in the open positive orthant (i.e., has all its components positive)? >> For n > 1 couldn't we have A = nxn identity, and v = (1,0,...,0) so that for no N is A^N v in the strictly positive orthant? Or is this like the Busy Beaver number -- we're only considering A's & v's such that A^N v is in the strictly positive orthant for *some* N ? << This can be recast as a purely combinatorial question. Suppose a finite directed graph with n vertices has the property that for every pair of vertices v,w, there is a way to get from v to w in N moves. Define the "inconvenience" of the digraph as the smallest N that works for all v,w. How large can the inconvenience of an n-vertex directed graph be? (One must assume that the digraph has finite inconvenience to begin with.) . . . >> I am a wee bit puzzled. Suppose we have a digraph with nodes {A,B,C,D} and arrows AB, BC, AD. Is its inconvenience = 2 ? (But there is no way to get from A to D in 2 moves, unless a move is allowed to be not moving.) --Dan

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[math-fun] inconvenient directed graphs
by James Propp 25 Sep '06

25 Sep '06

This must be in the literature, but I wasn't able to find it with Google: For each positive integer n > 0, what is the maximum value of N for which there exists an n-by-n stochastic matrix A and a probability vector v such that A^N v is the first forward image of v under repeated multiplication by A that lies in the open positive orthant (i.e., has all its components positive)? This can be recast as a purely combinatorial question. Suppose a finite directed graph with n vertices has the property that for every pair of vertices v,w, there is a way to get from v to w in N moves. Define the "inconvenience" of the digraph as the smallest N that works for all v,w. How large can the inconvenience of an n-vertex directed graph be? (One must assume that the digraph has finite inconvenience to begin with.) I can prove that the answer lies somewhere between quadratic-in-n and exponential-in-n, and I suspect that the former is where the truth lies. But my particular guess for quadratic truth must be sub-optimal, because the associated integer sequence isn't in Sloane. Jim Propp

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[math-fun] Brother Broussard type question:
by R. William Gosper 24 Sep '06

24 Sep '06

What is the least x>0 for which f(x):=prod(cos(x/n),n,1,inf) has a triple root? Putnam(?) version: what is the least c>0 for which lim c^x f(x) does not exist? x->oo --rwg Cowabunga Shredhead! (INCONSIDERATE CINOSTERNIDAE)

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Re: [math-fun] Most Transcendental
by Daniel Asimov 21 Sep '06

21 Sep '06

Kerry writes: << I know that a real number is either transcendental or it isn't, but is there a "most transcendental" number, in some sense like phi (1.618...) is the "most irrational" number? >> Roth's Theorem (cf. MathWorld) says this: All algebraic numbers x satisfy, for any eps > 0, | x - p/q | < 1/q^(2+eps) for only finitely many rationals p/q. This led to an irrationality measure (cf. MathWorld again) due to I'm not sure whom, as follows: ----------------------------------------------------------------------- Given x real, define irr(x) := inf over all u such that (*) |x - p/q| < 1/q^(u+eps) for all eps > 0 and for only finitely many rationals p/q. Quoting from < http://mathworld.wolfram.com/IrrationalityMeasure.html >: << mu(x) = = 1 if x is rational; = 2 if x is algebraic of degree > 1; >= 2 if x is transcendental. . . . >> This article includes some computed values (or bounds) for mu(x) for some familiar x's: (The table looks terrible to me in Arial (my only earthLlink webmail font choice), but it was created in Courier New, and I hope that font still makes it look good.) | | x | mu(x) >= | mu(x) <= | | ---------------------------------------------- | | e | 2 | 2 | | | | L | oo | oo | | | | pi | 2 | 8.0161 | | | | pi^2 | 2 | 5.4412 | | | | ln(2) | 2 | 3.8913 | | | | zeta(3) | 2 | 5.5138 ---------------------------------------------- where L := Liouville constant = sum_n=1^oo}1/10^(n!). (Note: mu(L) = oo means that there is no u for which (*) is true.) SO . . . mu(x) may be a good measure of "how transcendental" x is. --Dan

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[math-fun] Most Transcendental
by Kerry Mitchell 21 Sep '06

21 Sep '06

I know that a real number is either transcendental or it isn't, but is there a "most transcendental" number, in some sense like phi (1.618...) is the "most irrational" number? Kerry -- lkmitch(a)gmail.com www.fractalus.com/kerry

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[math-fun] New Mersenne prime, 2^32,582,657 - 1
by Schroeppel, Richard 21 Sep '06

21 Sep '06

A new Mersenne prime has been found, with exponent 32,582,657. This is the 44th known MP. It was found by the same team at Central Missouri State that found #43. A few details are at mersenne.org. The last few discoveries have been more closely bunched together than usual, and the new MP is still a little short of 10 million digits. Since the GIMPS project doesn't check exponents sequentially, there could be another MP with a smaller exponent, although this is unlikely. The double-checking thread of the work (which double checks the Lucas residues of the Mersenne composites) is lagging well behind. The last time I looked, it was working on numbers between 15M and 20M. Rich

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RE: [math-fun] Decimal question
by Eric Angelini 21 Sep '06

21 Sep '06

> 2340819/576 = 4063.921875 ... magnifique, Christian ! Best, É.

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[math-fun] Meta question: subject line tags
by Bernie Cosell 21 Sep '06

21 Sep '06

As many of you know, I'm no fan whatsover of those [LIST] subject line tags -- my email client handles that kind of info just fine for me, but folk seem to like the [go figure..:o)] and so I tolerate. Recently I've been trying to figure out why the threading is messed up on some threads here on math-fun and I just saw the culprit: the location of the subject tag on [some] replies. Most replies to the list are properly tagged: Subject: Re: [math-fun] Skolem-like primes sequence with the 'Re' at the beginning of the subject line, with the rest of the subject line intact [as per the RFC]. BUT.. several of the replies on this thread have the subject line: Subject: [math-fun] Re: Skolem-like primes sequence Which is *NOT* correct [the RFC says that the 'Re' on replies should come at the *beginning* of the subject line]. And so my [RFC compliant] email client correctly decides that that's the start of a separate thread and breaks the threading. Anyone know how that happens [and what if anything, can be done to make it stop]??? /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] Re: Meta question: subject line tags
by Mark D. Niemiec 19 Sep '06

19 Sep '06

"Stan E. Isaacs" <stan(a)isaacs.com> wrote: > But two different headings from the same responder suggest it > might not be coming from the mail client; perhaps Frank or Richard > could suggest the difference. Since I receive this as a digest, any attempt to respond begins by default with: Subject: Re: math-fun Digest, Vol 43, Issue 14 so I must always manually edit the subject line to respond to a specific message. Also, mailing list software typically adds the name of the list to the start of the subject line (unless it is already there). So, if I reply to a message as it originally appeared, rather than as the server previously edited it, it appears as: Subject: [math-fun] Re: Meta question: subject line tags rather than Subject: Re: [math-fun] Meta question: subject line tags What is the preferred protocol? -- Mark D. Niemiec <mniemiec(a)interserv.com>

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