Thanks for reply.
sorry, I made a mistake in reporting the information. Redid everything again. LibreOffice now shows a square image. When open the menu 'Format  image ', it shows 17cm by 17cm (the image in LibreOffice is reduced to fit the margins).
Output of "identify -verbose out.png":
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  Image: out.png
  Format: PNG (Portable Network Graphics)
  Mime type: image/png
  Class: DirectClass
  Geometry: 969x969+0+0
  Resolution: 600x600
  Print size: 1.615x1.615
  Units: Undefined
  Type: Bilevel
  Endianess: Undefined
  Colorspace: Gray
  Depth: 8/4-bit
  Channel depth:
    gray: 1-bit
    alpha: 4-bit
  Channel statistics:
    Pixels: 938961
    Gray:
      min: 0 (0)
      max: 255 (1)
      mean: 246.274 (0.96578)
      standard deviation: 46.3573 (0.181793)
      kurtosis: 24.2583
      skewness: -5.12429
    Alpha:
      min: 0 (0)
      max: 255 (1)
      mean: 7.70069 (0.0301988)
      standard deviation: 42.5589 (0.166898)
      kurtosis: 28.4677
      skewness: -5.48746
  Alpha: graya(255,0)   #FFFFFF00
  Colors: 16
  Histogram:
     24682: (  0,  0,  0,255) #000000 graya(0,1)
      1083: (  0,  0,  0,102) #00000066 graya(0,0.4)
       892: (  0,  0,  0,136) #00000088 graya(0,0.533333)
       694: (  0,  0,  0,187) #000000BB graya(0,0.733333)
       667: (  0,  0,  0, 68) #00000044 graya(0,0.266667)
       625: (  0,  0,  0, 17) #00000011 graya(0,0.0666667)
       608: (  0,  0,  0,238) #000000EE graya(0,0.933333)
       441: (  0,  0,  0,153) #00000099 graya(0,0.6)
       418: (  0,  0,  0,204) #000000CC graya(0,0.8)
       405: (  0,  0,  0, 51) #00000033 graya(0,0.2)
       404: (  0,  0,  0, 34) #00000022 graya(0,0.133333)
       382: (  0,  0,  0,221) #000000DD graya(0,0.866667)
       293: (  0,  0,  0, 85) #00000055 graya(0,0.333333)
       277: (  0,  0,  0,170) #000000AA graya(0,0.666667)
       260: (  0,  0,  0,119) #00000077 graya(0,0.466667)
    906830: (255,255,255,  0) #FFFFFF00 graya(255,0)
  Rendering intent: Undefined
  Gamma: 1
  Background color: graya(255,1)
  Border color: graya(223,1)
  Matte color: graya(189,1)
  Transparent color: graya(0,0)
  Interlace: None
  Intensity: Rec709Luminance
  Compose: Over
  Page geometry: 969x969+0+0
  Dispose: Undefined
  Iterations: 0
  Compression: Zip
  Orientation: Undefined
  Properties:
    date:create: 2015-08-14T17:49:54-03:00
    date:modify: 2015-08-14T17:49:54-03:00
    icc:copyright: Copyright Artifex Software 2011
    icc:description: Artifex Software sRGB ICC Profile
    icc:manufacturer: Artifex Software sRGB ICC Profile
    icc:model: Artifex Software sRGB ICC Profile
    pdf:Version: PDF-1.5 
    png:bKGD: chunk was found (see Background color, above)
    png:cHRM: chunk was found (see Chromaticity, above)
    png:iCCP: chunk was found
    png:IHDR.bit-depth-orig: 8
    png:IHDR.bit_depth: 8
    png:IHDR.color-type-orig: 4
    png:IHDR.color_type: 4 (GrayAlpha)
    png:IHDR.interlace_method: 0 (Not interlaced)
    png:IHDR.width,height: 969, 969
    png:pHYs: x_res=600, y_res=600, units=0
    png:text: 3 tEXt/zTXt/iTXt chunks were found
    png:tIME: 2015-08-14T20:49:54Z
    signature: 9e7290d5b7003c39d9c9676fb371bb99048421acb0b357ce9fcb841c4d9e28c1
  Profiles:
    Profile-icc: 2576 bytes
  Artifacts:
    filename: out.png
    verbose: true
  Tainted: False
  Filesize: 23.3KB
  Number pixels: 939K
  Pixels per second: 31.3MB
  User time: 0.040u
  Elapsed time: 0:01.030
  Version: ImageMagick 6.8.9-9 Q16 x86_64 2015-01-06 http://www.imagemagick.org
Link for original in.pdf (this is a mathematical image, with a circle. This is a test image): 
http://mnetox.atwebpages.com/in.pdf.
My Ghostscript version:
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GPL Ghostscript 9.15 (2014-09-22)
Copyright (C) 2014 Artifex Software, Inc.  All rights reserved.
Regarding the units used ...
Really I do not know what is the real meaning of the output of the identify command. He does not say the units used in the output.
What I do know are the units I'm using at the input: the original have PDF 4.1cm x 4.1cm (1.6in x 1.6in).
In the command below, I have used pixels and inches ('m new to ImageMagick and not fully understand the functioning of dpis):
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convert -density 600 -size 960 in.pdf out.png
Thanks.