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Testing Stellarium for angular measure:
  I find a minimum distance of 3.5651° or 12834 seconds of arc, which agrees with Antoine, Dave W, and Robert's revised value.

The long way to get distances, and the best way under most circumstances, is to collect coordinates for the event and then run a standard great circle calculation. But there's a way to do this without calculation.

Stellarium has an angular measure tool. Generally this is used for crude angular distances, but there's a way to get high accuracy from it. In this case, I go to the date and time in question. The time, by the way, only needs to be precise to the nearest few minutes since this is an angular minimum with a fairly long period with no significant change. I zoom in on Mars. I double-click on the visual center of Mars, happens to be on the edge of Olympus Mons (the giant volcano) at this time, and I drag out a short distance. Now I do a ctrl-F to find and jump to Venus. I right-click on the visual center of Venus (this is the least accurate part of this whole process). This "pulls" the end of the angle-measurement arc out to the center of Venus. Now ctrl-F to go back to Mars and zoom out a bit to see the resulting angle.

Note that you have to turn off the atmosphere in Stellarium (to eliminate refractions) and set it to show geocentric positions (turn off "topocentric" under Configuration > Tools). I also set the display to "decimal degrees" in the same tools dialog, but that's optional. Repeating the process a few times, I find that I can measure angles reliably (to the extent of the reliability of Stellarium itself!) to the nearest 0.0001° which is 0.36 seconds of arc, and of course that's at least ten times better than anything we might want for even the most extreme cases in celestial navigation work.

The difference between Robert's original distance and the revised distance might be due to inadvertently selecting topocentric coordinates in SOLEX. Although Mars on this date is over 2 AUs from the Earth yielding a very low topocentric parallax, Venus is 0.5 AUs from Earth so its parallax in altitude could be as large 18 seconds of arc.  Of course, in practice, for observation, we would want the parallax to be included, and refraction, too. For practical observations of this conjunction, we might also worry about the "center of light" correction which for Venus is also rather large on this date.

Antoine you referred to SOLEX as "Solar Experiment" in your post. Aldo Vitagliano, the author, just calls it "Solex" in all his docs. Do you know, was this the origin of the name SOLEX? I exchanged a few messages with Aldo Vitagliano a few years ago. Apart from maintenance work on the project, he is apparently no longer working on it (or at least he wasn't a couple of years ago). It's a very fine ephemeris product which I have been using for at least 15 years, but it suffers from an "unfortunate" interface. Of course, one can get away with that in a specialty product like SOLEX :).

Frank Reed