Paul Erdos
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Paul Erdos (1913-1996) was a prominent 20th century mathematician. Homeschooled as a youth, he ultimately published more math papers (1500) than anyone and developed an elementary proof for some of the most challenging math problems, including the the Prime Number Theorem and Bertrand's conjecture that there was always at least one prime between n and 2n for n > 2.
Unlike many 20th century mathematicians, Erdos adhered to using elementary techniques rather than using newly developed fields of mathematics. Erdos also focused on solving existing problems rather than creating new ones.