A Central History of the Prime Number Theorem

1) Numbers were used for record-keeping and commercial transactions for centuries before anyone thought ____ speculating ______ the nature and properties of the numbers themselves.

2) Around 600 B.C. Pythagoras and his disciples aimed _______ sorting out the positive integers in various ways; examples include even numbers, prime numbers, composite numbers.

3) Around 300 B.C. Euclid was engaged _______ writing his Elements, a remarkable collection of 13 books that contained much of mathematics known at that time.

4) Books 7, 8, 9 dealt _______ describing properties of the integers, which constitute the early beginnings of number theory.

5) Number theory has kept _______ drawing the attention of both amateurs and professional mathematicians for centuries.

6) The amateurs are capable _______ stating and understanding problems of the number theory.

7) The professionals are fond _______revealing relations of great depth and elegance derived from these same problems.

8) Proposition 20 in Book 9 of the Elements states that there are infinitely many primes. Many mathematicians succeeded ______ proving it but Euclid’s original proof is still the most elegant.

9) In 1737 Euler showed that adding the reciprocals of successive prime numbers (1/2+1/3+1/5+1/7+…) results _______ attaining a sum greater than any prescribed number.

10) Early attempts to prove the prime number theorem aided _______ developing the theory of functions of a complex variable – a branch of mathematics that is the lifeblood of mathematical analysis.

11) Efforts to prove Fermat’s last theorem account _______ arising algebraic number theory.

12) One unexpected application of algebraic number theory consists _______ designing security system for computers.