Here is the code that handles elgg registration http://reference.elgg.org/users_8php_source.html#l00919

i need the code in the form of a script which can be used in a directory away from the elgg site, in other words, a function or include call from a page on another website, but not an elgg site.and i need it without the line numbers.

Maybe this plugin is for you: http://community.elgg.org/plugins/1504035/0.1/embedded-login

no, that is a login, not registration. 

Try:

<iframe src='http://www.yoursite.com/register' style=' border-width:0 ' width='980'  height='1000' frameborder='0' scrolling='auto'></iframe>

i need the user of my site A (non-elgg) to login and click a link.

the link would activate code which would extract the user's details from the site A database and insert that data into my site B (elgg) database.

so framing the registration form will not work.

neither will paying [a dev shop] 450USD to do the job. (yes folks, thats what i have bern quoted!)

[Edited to remove personally identifying information.]

I have implemented something similar to what you may need.  

The web services api by Mark Harding should give you the baseline you need.

I would say is that you will need a developer to implement this for you because the API call will need to be done via REST, JSON or XML.  You will probably need make two calls one to authenticate as an admin and the other to atomically register.

In addition I think you may need to extend the registration API slightly depending upon your data requirements and also what happens when the users go directly to the Elgg site.  How would the end users successfully login?

What kind of system is site A running?

During login to site A (the only time you'd have access to the plaintext password) you'd need to store a 2nd hashed copy of it with Elgg's hashing algorithm. After registering the Elgg user (web services or however) you could then copy that hash and salt back into Elgg's user table, allowing the user to log in directly to Elgg.

No offense but you should not make privately quoted mails public. And, $450 is not an absurd price for the job.

Just a friendly advice. :)

All the very best for your project.

No offense but you should not make privately quoted mails public. And, $450 is not an absurd price for the job.

I agree, if you don't like the quote just move on.