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**Extra resources for Elementary Number Theory (Math 780 instructors notes)**

**Example text**

O 1+X1+ X 1+ = x ? xp + p1

X p1x ; log 1 + x1 = O(1=x): Comment: The expression O(g(x)) in an equation represents a function f (x) = O(g(x)). To clarify, the last equation in Xx X Xx X x = + O 1 = + O(x) p xp p x p xp p x p does not assert that a function is O function f (x) that satis es f (x) = O X p x 1 if and only if it is O(x) but rather there is a X p x 1 and f (x) = O(x). Indeed, in the equation above, the big oh expressions both represent the same function f (x) = An estimate using integrals. x . p p log x. (1) Let f : + !

P a twin prime p X Homework: (1) Let pn denote the nth prime. (a) Explain why the Prime Number Theorem implies that lim sup(pn+1 ? pn ) = 1. 1 (b) Use Theorem 43 to prove that for every positive integer k, ? lim sup minfpn+1 ? pn ; pn+2 ? pn+1 ; : : : ; pn+k ?