February 2020 - mathfun@mailman.xmission.com

[math-fun] Digitsum and Digitroot
by Éric Angelini 03 Mar '20

03 Mar '20

Hello Math-Fun, are 29, 38, 47, 48 and 49 the first five integers whose digitroot is NOT a substring of the digitsum? Best, É. Catapulté de mon aPhone

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[math-fun] In Memoriam: Freeman Dyson
by Brad Klee 02 Mar '20

02 Mar '20

The population size of Homo sapiens has gotten so large now, it is stating the obvious to say that, despite his fame, many people never had the opportunity to meet Freeman Dyson, to attend one of his talks or lectures, or to write letters with him. A smaller segment of the population, who only encountered his works in print or video-recorded talks, probably have a similar sadness and feeling of regret for a missed opportunity, say, to have a chat about the relative merits of hooliganism and the extinction rebellion. In this situation, we have to thank science for creating such detailed systems of classification that we can have a half-way valid, rational explanation for life stories that happen, and others that do not. Personally, I can still be happy having never met Freeman Dyson, and thoroughly enjoyed studying his theory for the life of the mind. The essay "The Scientist as a Rebel" [1] has been a huge influence in my most recent years. From his book by the same title, I particularly liked "A modern Heretic", a short essay on the thinking of Thomas Gold. These works are somewhat relevant to my postings on [seqfan] and [math-fun], because I wholeheartedly bought into the idea that professional training is not a deciding factor for performance in mathematical competitions. Of course the same is true for contests in physics, and Freeman Dyson himself, is a great example and inspirational personage as to what a trained mathematician can hope to accomplish in a neighbouring discipline. I also found interest in the video interview [2] at web of stories, enjoyed hearing the stories about Bethe, and think that Gosper has a funny story about the Ramanujan conference at UIUC. If anyone else has suggested reading or interesting stories to share, perhaps consider posting to the list or sending me a personal message. The fact that he is now deceased is no reason to stop studying his works, quite the opposite. --Brad [1] https://www.jstor.org/stable/2974452?seq=1 [2] https://www.webofstories.com/play/freeman.dyson/95

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[math-fun] Euler quote (?)
by James Propp 01 Mar '20

01 Mar '20

I just Googled the quote “*My pen is a better mathematician than I am*” (which I’m sure I’ve seen attributed to Euler) in search of the original French or Latin phrasing, and was dismayed to see that the only verbatim appearances of this sentence are references to an earlier essay of mine! I’m hoping this quote isn’t an unintentional fabrication on my part. Can anyone provide a source or reference? Thanks, Jim Propp

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[math-fun] Fibonacci, cosh, sinh
by Henry Baker 01 Mar '20

01 Mar '20

It is well-known that fib(n) = cosh(n*alpha)/cosh(alpha) (n odd) fib(n) = sinh(n*alpha)/cosh(alpha) (n even) alpha = asinh(1/2). Also, fib(n) = phi^n-(-phi)^(-n) ----------------- sqrt(5) Is there any version of these equations fib(n) = A*exp(a*n)+B*exp(b*n)+C*exp(c*n)+... that *doesn't* require splitting up into n odd, n even? (I'm willing to accept complex A,a,B,b,C,c, etc.)

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Re: [math-fun] sum of 2 triangular numbers is a triangular number
by Neil Sloane 28 Feb '20

28 Feb '20

I mentioned this on Feb 19 2020 and several people sent very helpful replies. For squares, we have the theorem that all solutions to x^2+y^2=z^2 in integers are given by Pythagorean triples, t*[u^2-v^2, 2uv, u^2+v^2]. Expanding on the replies from Michael Collins and Rich, it looks like the following is the analog for triangular numbers T_n = n(n+1)/2. ALL solutions to T_n+T_k=T_m are given by the list of what one might call Trithagorean triples: these are the triples [n,k,m] = [n, T/Q-(Q+1)/2,T/Q+(Q-1)/2] where n >= 2, T=n(n+1)/2, and Q is any odd divisor of T less than n, plus these triples with the first two coordinates swapped. I haven't found this in the literature, but it can hardly be new.

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Re: [math-fun] A + B + C = D and A, B, C substrings of D
by Tom Duff 25 Feb '20

25 Feb '20

I don't understand how you add up 1820 from 3 of its substrings. The available substrings are 1 2 8 18 20 82 182 820. one of them has to be as big as 1820/3=606 -- that's 820. One of the 2 remaining has to be as big as (1820-820)/2=500. That's also 820. The third has to be 1280-820-820=180, which isn't one of the choices. On Tue, Feb 25, 2020 at 12:10 PM Michael Greenwald <mbgreen(a)seas.upenn.edu> wrote: > > On 2020-02-25 10:28, Tom Duff wrote: > > I should say, I excluded 0 as an addend, because then you get a=a+0+0 > > for every a with a zero in it, which strikes me as not interesting. > > Adding 0's to a previous solution is also uninteresting. I assume > you meant to cut those out, too, but you (mistakenly?) included > all forms of 150* = 50* + 50* + 50* (but almost no other examples > [e.g. 182, 1820, 18200, ...], although you do include > 195000=5000+95000+95000 and 19500, and 195 = 5 + 95 + 95)

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[math-fun] a(n) and a(n+1) substrings of their product
by Éric Angelini 25 Feb '20

25 Feb '20

Hello Math-Fun, Carole Dubois and myself have tried to extend S -- but could not find a new term < 9 999 999 999. S = 1, 10, 11, 100, 21, 1000, 31, 10000, 41, 100000, 51, 1000000, 61, 10000000, 71, 100000000, 81, 2817. S should be the lexicographically earliest seq of distinct positive terms such that a(n) and a(n+1) are substrings of their product. Example: 81 * 2817 = 228177. Does S end there? Would a different a(1) produce more terms? Is there another way to exploit this idea of two terms visible in their product? (respectively addition, division) Or is this old hat? Best, É.

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[math-fun] Arkansas Events in March.
by Brad Klee 24 Feb '20

24 Feb '20

Flyer: https://0x0.st/iNl4.pdf . Thanks to the Arkansas Arthropod museum, we are also starting to get better entomological data, see for example: https://0x0.st/iNIj.jpg (the genus Phanaeus is an incredibly interesting case study for evolution, having travelled all across the Americas, into 50+ species). --Brad

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[math-fun] Progress in k-imperfect numbers
by Allan Wechsler 23 Feb '20

23 Feb '20

I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research. The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691. The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western Number Theory problem 99:08". (See Myerson's compendium of WNT problems at https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .) Douglas Iannucci made some progress on the problem in his 2006 paper -- see http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature. Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect. The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1) + ... +/- 1; the sign of the trailing unit depends on the parity of e. Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones. While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon. The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is 1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000. Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation. I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned.

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[math-fun] a formula for primes
by Simon Plouffe 23 Feb '20

23 Feb '20

Hello, Here is the english version of the article about primes : http://plouffe.fr/NEW/a%20formula%20for%20primes.pdf Best regards Simon Plouffe

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