[$]Recent-update-has-broken-the-dollar-sign[/$] - Page 4

Batalov · 2016-03-06, 22:27

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.

Nick · 2016-10-08, 08:59

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?

Dubslow · 2016-10-08, 10:10

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.

science_man_88 · 2016-10-08, 11:30

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.

Nick · 2016-10-08, 11:40

Is the following the problem (and solution)?

http://stackoverflow.com/questions/3...-vertical-line

henryzz · 2016-10-08, 18:17

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.

GP2 · 2016-10-08, 20:19

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

0PolarBearsHere · 2016-10-09, 08:52

Yep, safari sees it too.
Fixed in current mathjax based on a test at https://cdn.mathjax.org/mathjax/late...e-dynamic.html

LaurV · 2016-10-09, 11:37

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?

Xyzzy · 2016-10-09, 13:27

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!'"
  }
}

Dubslow · 2016-10-09, 14:04

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 <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.

2016-10-08, 10:10	#36
Dubslow Basketry That Evening! "Bunslow the Bold" Jun 2011 40<A<43 -89<O<-88 1C35₁₆ Posts	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. Attached Thumbnails

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2016-03-06, 22:27	#34
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 22406₈ 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, 11:40	#38
Nick Dec 2012 The Netherlands 2×23×37 Posts	Is the following the problem (and solution)? http://stackoverflow.com/questions/3...-vertical-line

2016-10-09, 08:52	#41
0PolarBearsHere Oct 2015 2×7×19 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 22614₈ 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?