Kudos To Wolfram Research

Ideas
1

I’d like to say congratulations to Stephen Wolfram and all the people at Wolfram Research for a great job with Mathematica 10.1.0. Mathematica 10 had a bumpy start but the major snags appear to have all been fixed. (It now loads fast in Windows, spell checking works, Save As goes to the proper directory,)

I’ve been using Mathematica since just before Mathematica 3 and there seem to have always been things to gripe about. Mathematica 6 was almost like an entirely new paradigm that threw many users off their stance. After that there was a lot experimentation on how to make Mathematica documents available to readers without Mathematica. WRI now seems to have a firm hold on that, although I’ve only barely explored CDF files and Cloud computing not at all.

WRI appears to have done work on improving their Function help pages. For example look at an old routine like Circle and you now see small diagrams clarifying the usage. Similarly, they also pay more attention to lower level functionality, so for example with MeshRegions it’s possible to extract all the simple lower level graphics primitives.

I like to use Mathematica by writing in notebooks as if writing on a blank sheet of paper. I don’t like the NewInsertionCell icon, or the various automatic pop-ups. But it’s possible to get rid of them with style sheets and options so it’s possible to work with that paradigm. One can still get entry completion by using Ctrl+K or Crtl+Shift+K. (I programmed one of my Logitech mouse buttons to initiate a Text cell, and since those along with Input cells are the most common in my usage, writing is very convenient.)

Mathematica 10.1 is a great product! If you’re new to Mathematica or contemplating using it there is a lot to learn, but you can have some confidence that your time investment will pay off.

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2

I am in total agreement with David.

I hope that this is the start of incremental updates (10.2, 10.3,…) for Mathematica. Getting new functionality (and bug fixes) in smaller, easy to digest doses is a good thing.

The main “major” upgrade I am waiting for now is the Cocoa (64 bit) front end for the Mac OS. I hope that I do not have to wait for version 11 for this.

7 Likes
3

Have you seen this, David?

In[1]:= Reduce[x^2 - 7 y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]
Out[1]= C[1] \[Element] Integers && C[1] >= 1 && 
 x == 1/2 ((8 - 3 Sqrt[7])^C[1] + (8 + 3 Sqrt[7])^C[1]) && 
 y == -(((8 - 3 Sqrt[7])^C[1] - (8 + 3 Sqrt[7])^C[1])/(2 Sqrt[7]))

let’s define x and y as functions of the exponent

In[2]:= Clear[x, y]
x[n_] := 1/2 ((8 - 3 Sqrt[7])^n + (8 + 3 Sqrt[7])^n)
y[n_] := -(((8 - 3 Sqrt[7])^n - (8 + 3 Sqrt[7])^n)/(2 Sqrt[7]))

and convince ourself that they are integers:

In[5]:= RootApproximant[x[#]] & /@ RandomInteger[{1, 30}, 17]

Out[5]= {32257, 514088, 9156316745224513962640064643072, \
2199950293420883836928, 2081028097, 8193151, 35061166466652774072320, \
33165873224, 130576328, 138038228081368154112, 543466014742176320, \
8193151, 8424001222568, 33165873224, 8, 138038228081368154112, \
574522862194843517270624305152}

In[6]:= RootApproximant[y[#]] & /@ RandomInteger[{1, 30}, 17]

Out[6]= {217149230841226888732894822400, 3096720, 50743789129440, \
13625260145284696589750763520, 12535521795, 49353213, \
223267627569198170583290140455600128, 12535521795, 12535521795, \
217149230841226888732894822400, 12888722657083742, \
53643587967663564981796864, 53643587967663564981796864, 
 1/4 (1617433305776647 + 13 Sqrt[15479825435713475684361539193]), 765, \
854931483328272233719136256, 223267627569198170583290140455600128}

In[7]:= FactorInteger[15479825435713475684361539193]
Out[7]= {{3, 1}, {7, 1}, {17, 1}, {173, 1}, {1160503, 1}, {215975989374144271, 1}}

they are not: the x pass this self-test, the y do not. This is under Mathematica 10.1. Mathematica is a great product, but as Heifetz put it:

There is no top. There are always further hights to reach.

Best possible support for Wolfram Inc. is pointing out the next bug.

1 Like
4

If you are interested in the exact values, try something like

 FullSimplify[Expand[y[#]]] & /@ RandomInteger[{1, 30}, 17]

I don’t think RootApproximant is under any obligation to return an integer, or even the same integer.

4 Likes
5

That works, you’re right

In[1]:= Clear[x, y]
x[n_] := 1/2 ((8 - 3 Sqrt[7])^n + (8 + 3 Sqrt[7])^n)
y[n_] := -(((8 - 3 Sqrt[7])^n - (8 + 3 Sqrt[7])^n)/(2 Sqrt[7]))

In[46]:= With[{l = RandomInteger[{1, 2^16}, 17]},
   Print["l: ", l];
   And @@ (IntegerQ /@ FullSimplify[Expand[x[#]] & /@ l])
 ]

During evaluation of In[46]:= l: {8309,55053,28081,62999,49730,51523,40250,65370,30882,4898,16978,48880,6337,6133,9837,26546,49300}
Out[46]= True

In[47]:= With[{l = RandomInteger[{1, 2^16}, 17]},
   Print["l: ", l];
   And @@ (IntegerQ /@ FullSimplify[Expand[y[#]] & /@ l])
]

During evaluation of In[47]:= l: {447,23621,25471,49476,26678,6573,60939,63208,56458,51985,24444,34525,59990,12914,10239,64048,34724}
Out[47]= True

thank you.

It has to work because in x the odd powers of \sqrt{7} cancel out as well in y the even powers of \sqrt{7} cancel out and in y a division by \sqrt{7} follows: x and y are both root-free. The factor \frac{1}{2} does also not create a rational number, because the previous step brought a factor 2 in the numerator.

1 Like
6

Cocoa front end would be very nice indeed, I find the front-end to be a little sluggish from time-to-time. This has become worst since Version 6 or so, and is most likely due to the increase of interactive features (popping up suggestions, hi-dpi 3D-graphics, 3d images,…)

1 Like