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Old 2016-03-06, 22:27   #34
Batalov
 
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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.
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Old 2016-10-08, 08:59   #35
Nick
 
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Quote:
Originally Posted by bgbeuning View Post
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?
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Old 2016-10-08, 10:10   #36
Dubslow
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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.
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Old 2016-10-08, 11:30   #37
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Quote:
Originally Posted by Nick View Post
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.
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Old 2016-10-08, 11:40   #38
Nick
 
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Is the following the problem (and solution)?

http://stackoverflow.com/questions/3...-vertical-line
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Old 2016-10-08, 18:17   #39
henryzz
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Quote:
Originally Posted by Dubslow View Post
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.
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Old 2016-10-08, 20:19   #40
GP2
 
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Quote:
Originally Posted by Nick View Post
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
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Old 2016-10-09, 08:52   #41
0PolarBearsHere
 
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Yep, safari sees it too.
Fixed in current mathjax based on a test at https://cdn.mathjax.org/mathjax/late...e-dynamic.html
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Old 2016-10-09, 11:37   #42
LaurV
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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?
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Old 2016-10-09, 13:27   #43
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Quote:
Originally Posted by GP2 View Post
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!'"
  }
}
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Old 2016-10-09, 14:04   #44
Dubslow
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Quote:
Originally Posted by Xyzzy View Post
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!'"
  }
}


A hard refresh (Ctrl+F5 or the like) may be required to flush the cache and fetch the new version.
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