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I know the minimum number of reference letters required at many U.S. PhD programs in the STEM fields is three -- and that it is also not uncommon for someone to submit a fourth letter.

What about a fifth letter? Would a fifth letter of recommendation be perceived as "too many"?

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    Have you considered the instructions on the application? Failure to follow simple instructions can be a big problem (even with egg-heads living a life of the mind). – Jacob Murray Wakem Sep 11 '16 at 3:28
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This can depend on the specific university department. Some may welcome it as supplemental material to your application whereas some may discourage excess letters and tell you to choose the ones that will best showcase your abilities as a student. Don't add things to your application that the department doesn't welcome.

I've generally heard and been told that an additional good recommendation letter will boost your chances of admission, but an additional generic recommendation letter that does not add anything unique to the other (assuming) 4 letters will only hurt it.

So for a general answer, make sure each recommendation letter is able to showcase different areas of skill rather than them overlapping with each other. Also, make sure to contact each department to make sure that they are okay with having more than 3 recommendation letters.

Hope this helps, good luck!

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Send exactly what was requested in the application instructions, and nothing else. Admissions committees suffer through reading all the application information, and do not appreciate getting extra things to read.

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    I want to disagree vehemently, having worked on admissions committees. Admissions committees like to accept the best students, and to do that, it helps to have as much information as possible. If a student is clearly going to be rejected, one additional letter of recommendation is not going to make the file take very much longer to read. If a student is under consideration for acceptance, and the additional letter provides relevant information not found in the other letters, the admissions committees I have been on would welcome it. – Peter Shor Sep 11 '16 at 16:05
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    Having said that, let me add that for many students, one of the three required letters of recommendation is generic and not useful for the admission committee. So I expect that it's quite rare for a student to have five recommenders who can all write a non-generic letter. But if you're that student, go ahead and submit five letters if it's allowed. – Peter Shor Sep 11 '16 at 16:18
  • @PeterShor Do your application instructions say "send at least three letters" or "send three letters"? Why? – Anonymous Physicist Sep 11 '16 at 23:47
  • The online form has three slots for recommenders. It also says "If you want more than 3 recommenders, please enter information for the first 3, save it, and return to this form." I have no idea if there's an upper limit. I also know that there have been applications where students didn't send in a letter from one of their research advisers, and we would have liked to see a letter from them. – Peter Shor Sep 12 '16 at 0:01
  • The admissions process at most universities is now online. If a departments really wants people to list only three recommenders, they can make it impossible to submit more. And if what you want to convey in your answer is "don't send more than three letters if the instructions say "send exactly three," that's just common sense. – Peter Shor Sep 12 '16 at 1:47

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