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 2016-03-06, 22:27 #34 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 41·229 Posts McDaniel, Wayne L. Perfect Gaussian integers. Acta Arith. 25 (1973/74), 137--144. 12A05 ----------------------------------------------------------------------------- Let $\eta=\varepsilon\prod\pi_i{}^{k_i}$ be a Gaussian integer, $\varepsilon$ a unit, $\pi_i$ primes, $\text{Re}\,\pi_i>0$, $\text{Im}\,\pi_i\geq 0$. R. Spira [Amer. Math. Monthly 68 (1961), 120--124; MR 26 #6101] defined the divisor sum function by means of $\sigma(\eta)=\prod(1+\pi_i+\cdots+\pi_i{}^{k_i})=\prod(\pi_i{}^{k_i+1}-1)/( \pi_i-1)$. The concepts even, odd, Mersenne prime, perfect numbers were extended as follows: (i) $\eta$ is an even Gaussian integer if $(1+i)|\eta$ and an odd integer if $(1+i)\not \mid\eta$. (ii) The sum $\sigma((1+i)^{k-1})=-i[(1+i)^k-1]=M_k$ is called a complex Mersenne prime if $M_k$ is a prime. (iii) $\eta$ is perfect if $\sigma(\eta)=(1+i)\eta$ and is norm-perfect if $\|\sigma(\eta)\|=2\|\eta\|$, where $\|\sigma(\eta)\|=\eta\eta^*$, the norm of $\eta$. If a (norm-) perfect number $\eta$ does not have a (norm-) perfect number as a proper divisor, then $\eta$ is primitive. The main result of the present paper is contained in the following theorem: Let $M_p$ be a complex Mersenne prime and $\varepsilon$ a unit. If $p\equiv 1 (\text{mod}\,8)$, $\eta=\varepsilon(1+i)^{p-1}M_p$ is a primitive norm-perfect number; if $p\equiv-1 (\text{mod}\,8)$, $\eta=\varepsilon(1+i)^{p-1}M_p{}^*$ is a primitive norm-perfect number. Conversely if $\eta$ is an even primitive norm-perfect number then, for some unit $\varepsilon$, either $\eta=\varepsilon(1+i)^{p-1}M_p$ and $p\equiv 1 (\text{mod}\,8)$ or $\eta=\varepsilon(1+i)^{p-1}M_p{}^*$ and $p\equiv-1 (\text{mod}\,8)$; in each case $M_p$ denotes a complex Mersenne prime. Corollary: $\eta$ is a primitive perfect number if and only if there exists a rational prime $p\equiv 1 (\text{mod}\,8)$ such that $\eta=(1+i)^{p-1}M_p$. The author gives $\eta=(1+i)^6(7+8i)^2(7+120i)$ as the simplest example of an imprimitive norm-perfect number. He also proves that if $\varepsilon$ is a unit, then (a) if $p\equiv 1 (\text{mod}\,8)$ and $M_p$ and $\sigma(M_p{}^2)$ are primes then $\eta=\varepsilon(1+i)^{p-1}M_p{}^2(\sigma(M_p{}^2))^*$ is an imprimitive norm-perfect number; (b) if $p\equiv-1 (\text{mod}\,8)$ and $M_p$ and $\sigma(M_p{}^{*2})$ are primes then $\eta=\varepsilon(1+i)^{p-1}M_p{}^{*2}(\sigma(M_p{}^{*2}))^*$ is an imprimitive norm-perfect number.
2016-10-08, 08:59   #35
Nick

Dec 2012
The Netherlands

13×127 Posts

Quote:
 Originally Posted by bgbeuning Have not seen the bar symbol used as a suffix operator before. Or is it a type setting thing. For me the first line displays as Let a(bar) and b(bar) be integers.
The attached image shows what I see.
Had we decided on a method for embedding mathematics which works for everyone, or is the discussion still going on?
Attached Thumbnails

 2016-10-08, 10:10 #36 Dubslow Basketry That Evening!     "Bunslow the Bold" Jun 2011 40
2016-10-08, 11:30   #37
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

203008 Posts

Quote:
 Originally Posted by Nick The attached image shows what I see. Had we decided on a method for embedding mathematics which works for everyone, or is the discussion still going on?
I'm not sure if the use of \ \ style versus the dollar sign tags given in the tags now might be part of it. $$\Sigma a$$ for example versus $$\Sigma a$$ $\Sigma a$ nope using [tex] tags makes Batalov's work on mine though.

 2016-10-08, 11:40 #38 Nick     Dec 2012 The Netherlands 13·127 Posts Is the following the problem (and solution)? http://stackoverflow.com/questions/3...-vertical-line
2016-10-08, 18:17   #39
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

3×1,951 Posts

Quote:
 Originally Posted by Dubslow This is what I see on my new operating system. A month ago on the prior one I had no such issues here. I can confirm that http://www.rechenkraft.net/aliquot/intro-analysis.html displays with equally exaggerated size, but no bars like here in the forum. I don't know what could be the cause. I don't think it's a configuration issue, but I suppose it could be.
I have been seeing that on and off.

2016-10-08, 20:19   #40
GP2

Sep 2003

13×199 Posts

Quote:
 Originally Posted by Nick Is the following the problem (and solution)? http://stackoverflow.com/questions/3...-vertical-line
Sure sounds like it, so the solution would be to upgrade to MathJax 2.6

Wikipedia says the latest stable release is 2.6.1

 2016-10-09, 08:52 #41 0PolarBearsHere     Oct 2015 1000010102 Posts Yep, safari sees it too. Fixed in current mathjax based on a test at https://cdn.mathjax.org/mathjax/late...e-dynamic.html
 2016-10-09, 11:37 #42 LaurV Romulan Interpreter     Jun 2011 Thailand 33·347 Posts Firefox renders it right, I see it exactly like in Nick's snap photo. If you right-click any equation (to open the Mathjax popup dialog), select "math settings" and then and select different "math rendering" options, doesn't that solve the problem?
2016-10-09, 13:27   #43
Xyzzy

"Mike"
Aug 2002

2·5·11·73 Posts

Quote:
 Originally Posted by GP2 Sure sounds like it, so the solution would be to upgrade to MathJax 2.6 Wikipedia says the latest stable release is 2.6.1
We think we have updated MathJax to the newest version. Let us know if this fixes the problem.

Code:
$cat package.json { "name": "mathjax", "version": "2.6.1", "description": "Beautiful math in all browsers. MathJax is an open-source JavaScript display engine for LaTeX, MathML, and AsciiMath notation that works in all browsers.", "keywords": [ "math", "svg", "mathml", "tex", "latex", "asciimath", "browser", "browser-only" ], "maintainers": [ "MathJax Consortium <info@mathjax.org> (http://www.mathjax.org)" ], "bugs": { "url": "http://github.com/mathjax/MathJax/issues" }, "license": "Apache-2.0", "repository": { "type": "git", "url": "git://github.com/mathjax/MathJax.git" }, "main": "./MathJax.js", "scripts": { "test": "echo 'No tests here!'" } } 2016-10-09, 14:04 #44 Dubslow Basketry That Evening! "Bunslow the Bold" Jun 2011 40<A<43 -89<O<-88 3·29·83 Posts Quote:  Originally Posted by Xyzzy We think we have updated MathJax to the newest version. Let us know if this fixes the problem. Code: $ cat package.json { "name": "mathjax", "version": "2.6.1", "description": "Beautiful math in all browsers. MathJax is an open-source JavaScript display engine for LaTeX, MathML, and AsciiMath notation that works in all browsers.", "keywords": [ "math", "svg", "mathml", "tex", "latex", "asciimath", "browser", "browser-only" ], "maintainers": [ "MathJax Consortium (http://www.mathjax.org)" ], "bugs": { "url": "http://github.com/mathjax/MathJax/issues" }, "license": "Apache-2.0", "repository": { "type": "git", "url": "git://github.com/mathjax/MathJax.git" }, "main": "./MathJax.js", "scripts": { "test": "echo 'No tests here!'" } }

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