Saratoga was the Reliant-class ship we saw at the beginning of the movie encountering the probe for the first time. 

Admiral Cartwright's sitrep to the Federation President has him calling up the Yorktown's report to Starfleet, in which that ship's captain mentions the solar sail attempt (as well as the amount of time that has past since crossing paths with the probe). 

True.  I'm not sure if the Yorktown was established as being renamed as the Enterprise in the film or after the films release.  Star Trek V gives a different impression the 1701-A was a new ship with shakedown operations at hand.  1701-A is a new star fleet vessel not a refit star ship.

Was it established in classic series which registry number is accurate for the Farragut?

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I know the previous DC Comics Trek adapations are not considered canon, but the "Debt of Honor" graphic novel revealed the USS Farragut to be a Constitution class ship with the registry NCC-1702.

Edited to remove unnecessary "Debt of Honor" info.

Uh, just checked my Star Trek Encyclopedia (1999 ed)...  The individual entry for USS Farragut is catagorizing her as a Constitution-Class with registry NCC-1647... Well, that opens up the forementioned can of blood worms.

We know the Repulic and the Farragut were operative sometime during
Spock's 13-year tour under Pike, since Kirk was on both ships. One could
extrapolate that they were contemporaries of the Enterprise in that sense.
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Although their classes were never stated, this seems reasonable.

As for Constellation, some fans have justified the lower registry number by saying that she was an older-class ship that was heavily upgraded to Constitution specs.

Exeter and Defiant could also have been contemporaries.

It's important to keep in mind that the registry numbers currently in vogue (which were never seen on-screen, mind you, save for Constellation--and the mention of Republic's) come from Greg Jein's fan essay from 1975.

http://www.trekplace.com/article10.html

The actual numbers come from the wall chart in "Court Martial". Jein just matched them up with the known Constitution-class ships. The numbers weren't really adopted by the fanbase as belonging to those ships until the publication of the Encyclopedia in 1993.

Prior to all of that, Franz Joseph's numbers (which had much more logical/sequential NCC-17** registries for the Constitution-class ships) were used for novels, reference books, and role-playing games.

Anyway, as it stands right now, the Yorktown seems like my most likely
choice, since we know she survived TOS and was even "refitted" a la the
Enterprise. The Yorktown is seen as an operating ship in THE
VOYAGE HOME, set long after the period of CREW.
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Yorktown was also the original name for Star Trek's ship, so that seems appropriate.

^^ Fascinating reading Greg, thanks for the link!

Correct, Al.

As for the Enterprise-A, Roddenberry suggested that it be a renamed Yorktown in honor of the fact that Yorktown was the name used in his original outline.

However, the only on-screen evidence to suggest that the ship was not newly-built was its surprisingly early retirement in Star Trek VI.

 I figured I'd post this question here rather than dedicating an entirely new thread to it:

     JB: I know that you are happy to be working on STAR TREK and satisfied with what you've produced so far, as well as with how well it's generally been received. I was curious, given your longstanding appreciation of the property, as well as the influence it has had on your other work, if working in the STAR TREK universe represents a long sought-after "dream gig" for you. Is this something you've always wanted to do, and how high up the list of projects that you've worked on does STAR TREK rank? How happy is that kid in a candy store compared to working on other pre-existing properties that you've had an affection for, like FANTASTIC FOUR?

However, the only on-screen evidence to suggest that the ship was not newly-built was its surprisingly early retirement in Star Trek VI.

***

Not if Star Fleet decided all Constitution vessels to be decommissioned to bring forth more state of the art Star Fleet vessels like Excelsior, Reliant, and Oberth class vessels.  Also the Enterprise could have been used as a vessel for interstellar travels for trainees.   

Another solution, suggested by a friend (Hi, Paul!), is that the registration
numbers of previous ships with the same name are carried over to new
ships, whatever their class. Thus, if there was an earlier Constellation,
the new ship would pick up the older registration number.

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Makes sense.

If  NX-01 had been renumbered NCC-01, then we'd wonder why Kirk's ship was designated NCC-1701...a tidbit which indicates that Enterprise got wiped from history somehow.

Jefferies' original idea for the "NCC-1701-A" number was that it would be applied to the first major modernization/upgrade of Kirk's ship (although the idea wasn't used in ST: TMP, but the number later appeared on a brand new Enterprise in ST IV).

For the sake of CREW, however, I
will, on the few occasions when such references are needed, rely on the
more logically numerical sequencing. My story, after all, takes place before
the first episode of TOS. Thus, the Enterprise is NCC-1701 as the
second ship of the new Class.

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As it should be, I think!

Guys, isn't it fun to talk about STAR TREK?

JJ's film is going to kill the experience for the teenyboppers because it will be just another summer romp like Transformers.  Brainless entertainment courtesy of Paramount Pictures!