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[math-fun] Dense unit distance graphs.
by Allan Wechsler 17 Sep '18

17 Sep '18

For a given set of k points in the plane, some number n of the mutual distances are 1. How big can we make n for a given k? A186705 at OEIS given the answers we know, but only up to 14 vertices. Where can I see pictures of example maximal unit distance graphs? The first "surprise" is a(9)=18; I would love to see that one.

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[math-fun] please let this *not* be pi
by Wouter Meeussen 15 Sep '18

15 Sep '18

what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry? for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i and so the running average up to n goes like n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i no sweet convergence sofar just curious, Wouter

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[math-fun] mersenne primes and their multiples
by Cris Moore 15 Sep '18

15 Sep '18

I was wondering what primes can occur as factors of “sparse” integers, i.e. those with only a few 1s in their binary expansion. Contrapositively speaking, you might enjoy proving the following: any multiple of the Mersenne prime 2^k-1 has at least k 1s. - Cris

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[math-fun] FromContinuedFraction@{0, 1, 1, 2, 2, 3, 3, 4, 4, . . .}
by Bill Gosper 15 Sep '18

15 Sep '18

Finch, p424 (Euler-Gompertz Constant): "No one knows the exact outcome if we instead repeat each denominator in c1 [ := FromContinuedFraction@{0, 1, 2, 3, 4,. . .} = I_1(2)/I_0(2)] ~ .6977746579640] although numerically we find that c2 = .5851972651050... ". Using my mysteriously wrong-but-correctible matrix-to-CF (non-regular) converter, c2= 1/2 + 1/(12 + 2*ContinuedFractionK[1 - k^2, 1 + 3*k + 2*k^2 + k^3, {k, 2, ∞}]). Julian & I had a way to further convert these to log-derivatives of pFqs, if anyone is really interested. --rwg

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[math-fun] Salem number primes
by Ed Pegg Jr 15 Sep '18

15 Sep '18

At http://www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html is a list of the 47 smallest known Salem number polynomial. Here's a list of 47 small numbers. {1, 1, 1, 1, 1, 1, 5, 11, 1, 3, 1, 7, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 13, 19, 5, 1, 1, 13, 5, 1, 7, 1, 1, 41, 1, 1, 3, 19, 1, 1, 1, 1, 5, 3, 7, 1} If you take the discriminant of the polynomial and divide by the corresponding prime the result is a perfect square number. --Ed Pegg Jr

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Re: [math-fun] mersenne primes and their multiples
by Keith F. Lynch 13 Sep '18

13 Sep '18

Cris Moore <moore(a)santafe.edu> wrote: > I was wondering what primes can occur as factors of ?sparse? integers, > i.e. those with only a few 1s in their binary expansion. How few is few? I looked at 21-bit numbers with three or fewer one-bits. Zero one-bits of course only gives me zero. One one-bit gives me powers of two, whose only prime factor is two. Two one-bits is where it gets interesting. I only looked at numbers whose rightmost bit is one, since all others are equal to such a number times some power of two. Here are the prime factors of the two-one-bit numbers: 3: 3 5: 5 9: 3^2 17: 17 33: 3 11 65: 5 13 129: 3 43 257: 257 513: 3^3 19 1025: 5^2 41 2049: 3 683 4097: 17 241 8193: 3 2731 16385: 5 29 113 32769: 3^2 11 331 65537: 65537 131073: 3 43691 262145: 5 13 37 109 524289: 3 174763 1048577: 17 61681 Every other one is divisible by three. Every fourth one is divisible by five. None are divisible by seven. Do all primes that ever appear appear at regular intervals? Here are the prime factors of the three-one-bit numbers: 7: 7 11: 11 13: 13 19: 19 21: 3 7 25: 5^2 35: 5 7 37: 37 41: 41 49: 7^2 67: 67 69: 3 23 73: 73 81: 3^4 97: 97 131: 131 133: 7 19 137: 137 145: 5 29 161: 7 23 193: 193 259: 7 37 261: 3^2 29 265: 5 53 273: 3 7 13 289: 17^2 321: 3 107 385: 5 7 11 515: 5 103 517: 11 47 521: 521 529: 23^2 545: 5 109 577: 577 641: 641 769: 769 1027: 13 79 1029: 3 7^3 1033: 1033 1041: 3 347 1057: 7 151 1089: 3^2 11^2 1153: 1153 1281: 3 7 61 1537: 29 53 2051: 7 293 2053: 2053 2057: 11^2 17 2065: 5 7 59 2081: 2081 2113: 2113 2177: 7 311 2305: 5 461 2561: 13 197 3073: 7 439 4099: 4099 4101: 3 1367 4105: 5 821 4113: 3^2 457 4129: 4129 4161: 3 19 73 4225: 5^2 13^2 4353: 3 1451 4609: 11 419 5121: 3^2 569 6145: 5 1229 8195: 5 11 149 8197: 7 1171 8201: 59 139 8209: 8209 8225: 5^2 7 47 8257: 23 359 8321: 53 157 8449: 7 17 71 8705: 5 1741 9217: 13 709 10241: 7^2 11 19 12289: 12289 16387: 7 2341 16389: 3^3 607 16393: 13^2 97 16401: 3 7 11 71 16417: 16417 16449: 3 5483 16513: 7^2 337 16641: 3^2 43^2 16897: 61 277 17409: 3 7 829 18433: 18433 20481: 3 6827 24577: 7 3511 32771: 32771 32773: 13 2521 32777: 73 449 32785: 5 79 83 32801: 32801 32833: 32833 32897: 67 491 33025: 5^2 1321 33281: 23 1447 33793: 47 719 34817: 37 941 36865: 5 73 101 40961: 40961 49153: 13 19 199 65539: 65539 65541: 3 7 3121 65545: 5 13109 65553: 3 21851 65569: 7 17 19 29 65601: 3^2 37 197 65665: 5 23 571 65793: 3 7 13 241 66049: 257^2 66561: 3 11 2017 67585: 5 7 1931 69633: 3^3 2579 73729: 17 4337 81921: 3 7 47 83 98305: 5 19661 131075: 5^2 7^2 107 131077: 23 41 139 131081: 19 6899 131089: 7 61 307 131105: 5 13 2017 131137: 71 1847 131201: 7 18743 131329: 11 11939 131585: 5 26317 132097: 7 113 167 133121: 133121 135169: 29 59 79 139265: 5 7 23 173 147457: 147457 163841: 163841 196609: 7 28087 262147: 262147 262149: 3 87383 262153: 262153 262161: 3^2 29129 262177: 23 11399 262209: 3 87403 262273: 11 113 211 262401: 3 47 1861 262657: 262657 263169: 3^6 19^2 264193: 349 757 266241: 3 88747 270337: 270337 278529: 3 227 409 294913: 41 7193 327681: 3^2 23 1583 393217: 11 35747 524291: 29 101 179 524293: 7 11^2 619 524297: 17 30841 524305: 5 19 5519 524321: 7 74903 524353: 524353 524417: 61 8597 524545: 5 7^2 2141 524801: 524801 525313: 525313 526337: 7 17 4423 528385: 5 11 13 739 532481: 647 823 540673: 7 77239 557057: 557057 589825: 5^2 23593 655361: 7 251 373 786433: 786433 1048579: 7 163 919 1048581: 3^2 263 443 1048585: 5 209717 1048593: 3 7 13 23 167 1048609: 1048609 1048641: 3 11 43 739 1048705: 5 7 19^2 83 1048833: 3^2 116537 1049089: 1049089 1049601: 3 7 151 331 1050625: 5^4 41^2 1052673: 3 350891 1056769: 7 150967 1064961: 3^3 39443 1081345: 5 23 9403 1114113: 3 7^2 11 13 53 1179649: 1179649 1310721: 3 67 6521 1572865: 5 7 44939 The first few primes that don't appear are 31, 89, and 127, two of which are Mersenne primes. (Well, okay, 2 never appears, but only because I skipped all even numbers.) > Contrapositively speaking, you might enjoy proving the following: > any multiple of the Mersenne prime 2^k-1 has at least k 1s. Every number in the form 2^k-1, whether prime or not, consists of k adjacent one-bits. Whenever you multiply such a number by anything, you are adding shifted versions of that string of k adjacent one-bits. Due to carries, that results in a sequence of one-bits over all of the overlap region except the rightmost bit of the overlap. You also get one one-bit for each bit to the right of the overlap. And due to carries, you get at least one one-bit to the left of the overlap. So that totals at least k one-bits.

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[math-fun] Binomial Coefficient Nomenclature
by Michael Collins 12 Sep '18

12 Sep '18

Is there any standard terminology -- analogous to the "numerator" and "denominator" of a fraction -- for the "n" and "k" in "n choose k"? If not, I'm sure people on this list could come up with some good suggestions ... -Mike Collins

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[math-fun] music and math
by Cris Moore 11 Sep '18

11 Sep '18

For your enjoyment: https://portal.knme.org/video/majesty-of-music-and-math-egpyyf/ <https://portal.knme.org/video/majesty-of-music-and-math-egpyyf/> - Cris

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Re: [math-fun] Secure internet (OT)
by Henry Baker 11 Sep '18

11 Sep '18

Hi Dan: If you have reasonably current Apple gear, and *if it is configured properly* (the default configuration for Apple gear is "usually" pretty good), and if you use reasonable passwords, then you should be reasonably secure. If you have Apple gear that is so old that it is no longer receiving updates, then get new gear that is, because so many vulnerabilities have been found just in the past year or so, that firmware updates are mandatory. Apple is better than most other vendors at trying to keep their customers "reasonably" secure. An antivirus (AV) program is a government surveillance expert's dream: if you hand it a "definition" file, it is a shopping list of files the AV program is constantly pawing through your system trying to find. So the easiest way for a hacker to find something on your computer is to hack the AV program and ask *it* to look for the stuff. AV programs also tend to get the highest access "privileges", so once they've been compromised, you're toast. Nowadays, the biggest problems aren't the gear or its software, but clicking on "phishing" emails. If you click on what you *thought* was an email from your bank or your internet service provider, you will be in big trouble. This is how the Democratic National Committee was hacked -- it wasn't due to any HW/SW vulnerability. The other problem with individual home networks these days is that it takes only *one* vulnerable device to be hacked for someone to gain a foothold into your home network. After that, the hacked device can bang away incessantly 24x7 until it finds additional vulnerabilities. For example, a "smart" TV, a "smart" speaker, a "smart" home security camera, a "smart" thermostat, a "smart" fitness device, etc., can all be hacked. If any of these devices ask for your wifi password, simply use Nancy Reagan's rule: "just say no". If you have kids, or kids want to use your wifi, consider getting a separate wifi router that is used *only* for guests; they just aren't that expensive anymore (I purchased one for $35 new that works just fine, but it isn't Apple). There is one excellent rule for home routers: disable "plug-and-play" (UPnP), and don't allow any other devices -- e.g., game consoles -- to turn it back on. UPnP has been involved in too many vulnerabilities. Make sure that you use "WPA2" for your wifi passwords, and use relatively long WPA2 passwords. So-called "WEP" is totally broken & can be hacked in mere minutes. WPA3 will be even better when it finally comes out, as it provides for better protection against brute force attacks by someone in your neighborhood, or in a parked van. You might consider changing your DNS provider to Google (8.8.8.8), rather than relying on your own ISP's DNS service. While this gives Google access to every DNS lookup, Google is probably less likely to give you a fraudulent IP address from such a lookup. You can change the DNS server not only in your browser(s), but also in your computer, and in your router. Another rule that may be difficult for many people to agree to: turn off "SMB" file access on your home network. The Microsoft "SMB" protocols have been an infinite source of vulnerabilities. Another rule: completely remove Adobe Flash from every device that you own. It has also been a nightmare of vulnerabilities. Tor is just fine; I use it every day for general browsing. Because of the Tor Project's paranoia, they've done a lot to try to keep someone from the internet from messing up your machine(s). However, I wouldn't use Tor for accessing your bank accounts, because that would force your traffic to go overseas and back, and eliminates the possibility that your bank's own routing safeguards will protect you. There have been a number of articles on using VPN's (Virtual Private Networks). If you choose to go this route(!), make sure that you trust your VPN provider implicitly, because 1) they can see (and potentially manipulate) everything; and 2) you've now made it impossible for many other safeguards to operate -- e.g., firewalls. You should find out your own current IP address (there are a number of web sites that will tell you what it is just now; it can change from day to day or even hour to hour). Then run a *port scan* against your own IP address. If *any port is open*, then you should find out what it is and close it. Unless you are running your own web site or email server, you shouldn't have any ports open onto the main internet. At 07:09 PM 9/9/2018, Dan Asimov wrote: >I've had at least two computer breaches in the past two years, and further >reason to suspect my internet setup is not secure. > >So I wonder if math-fun folks who know about such things can suggest to me how to >set up a (relatively!) secure home internet from scratch. > >I'm a Mac shop, so will need Apple-compatible stuff. > >My concerns include the following: > >1) I'd like the router security to be reasonably unbreakable; > >I'm not sure how to achieve this. > >2) I'd like my own internet use to be private and reasonably un-break-in-able; > >This would involve at least some security software; I've used MacScan in the past >without evident problems; I don't trust Norton or McAfee. > >Maybe it also would require the Tor browser or something like it? I've avoided this >so far because I don't want to find myself on some gov't watchlist just because I use it. > >3) I would also like to be protected from someone in a windowless van down the street, >containing special equipment, eavesdropping on my internet use. > >Humor me for a moment no matter how unlikely you may think this scenario. > >Is there a not-too-expensive way to reasonably protect oneself from that kind of >eavesdropping? (Or at least a way to catch and have prosecuted the perpetrators?) > >All helpful comments will be much appreciated! > >Dan

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[math-fun] Pi: acceleration of the speed of convergence
by françois mendzina essomba2 11 Sep '18

11 Sep '18

Hello, dear all Here are two formulas of Pi, the first is convergence 2 * n + 1 And the second 2 * n + 3 The formulas converge to an odd multiple of Pi depending on the x [0] taken, for example x[0]=3 ; x[m+1]=x[m]+sum(2*(n!)^2*sin(k*x[m])/(k*(n+k)!*(n-k)!),k=1..n); x[m+1]=x[m]+((2*n+1)!)^2/(2^(4*n-1)*(n!)^2)*sum(cos((2*k+1)*x[m]/2)/((2*k+1)^2*(n-k)!*(n+k+1)!),k=0..n); The two formulas are particular cases of thousands of formulas of this form of which I have been able to highlight. Their transformation into linear convergence can allow me to reach 500 decimal places in each calculation. Best Regards... FME...

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