2 Edited for clarity--I wasn't suggesting that these programs due conjecture/proof study of number theory, just that these programs often focus on number theory.
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As a number theorist I am biased, but I think a conjecture-proof study of elementary number theory could be a good place to start. Many

Many summer programs for mathematically-inclined high school students (e.g., the Ross Program, the Hampshire College Summer Studies in Math, PROMYS) focus on number theory and these have a long track record of producing mathematicians who go into academia. It

Doing number theory in this way combines many aspects of the other responses: (1) interesting problems with little technical background, (2) development of programming skills as you formulate and verify conjectures, (3) development of proof-writing and (4) applications (cryptography, mostly). Books such as

https://www.maa.org/press/ebooks/number-theory-through-inquiry

and

http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0470412151&bcsId=4878

might be good places to start.

As a number theorist I am biased, but I think a conjecture-proof study of elementary number theory could be a good place to start. Many summer programs for mathematically-inclined high school students (e.g., the Ross Program, the Hampshire College Summer Studies in Math, PROMYS) focus on number theory and these have a long track record of producing mathematicians who go into academia. It combines many aspects of the other responses: (1) interesting problems with little technical background, (2) development of programming skills as you formulate and verify conjectures, (3) development of proof-writing and (4) applications (cryptography, mostly). Books such as

https://www.maa.org/press/ebooks/number-theory-through-inquiry

and

http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0470412151&bcsId=4878

might be good places to start.

As a number theorist I am biased, but I think a conjecture-proof study of elementary number theory could be a good place to start.

Many summer programs for mathematically-inclined high school students (e.g., the Ross Program, the Hampshire College Summer Studies in Math, PROMYS) focus on number theory and these have a long track record of producing mathematicians who go into academia.

Doing number theory in this way combines many aspects of the other responses: (1) interesting problems with little technical background, (2) development of programming skills as you formulate and verify conjectures, (3) development of proof-writing and (4) applications (cryptography, mostly). Books such as

https://www.maa.org/press/ebooks/number-theory-through-inquiry

and

http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0470412151&bcsId=4878

might be good places to start.

1
source | link

As a number theorist I am biased, but I think a conjecture-proof study of elementary number theory could be a good place to start. Many summer programs for mathematically-inclined high school students (e.g., the Ross Program, the Hampshire College Summer Studies in Math, PROMYS) focus on number theory and these have a long track record of producing mathematicians who go into academia. It combines many aspects of the other responses: (1) interesting problems with little technical background, (2) development of programming skills as you formulate and verify conjectures, (3) development of proof-writing and (4) applications (cryptography, mostly). Books such as

https://www.maa.org/press/ebooks/number-theory-through-inquiry

and

http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0470412151&bcsId=4878

might be good places to start.