For J.King's Number Theory class,
 S00..(3)..MAS4203   3173.....Intro.Numb.Theor.....MWF5.....LIT217...King

  Here are the extra, spur of the moment, problems that Prof. King
somehow manages to come up with.  This list is maintained by

	Todd Behrens  and  Jason Thomas.

Abbrevs: Use #(A) for the cardinality of set A.
  Use oo for "infinity".  Use _ for subscript and ^ for superscript.
Use {} for grouping and also to indicate a set.  Use \in for the "is
an element of" symbol.
  Use \int for integral.  Thus
	\int_{R+2}^{R+5} x^3 dx
means "the integral, as x goes from R+2 to R+5, of x^3 times dx".

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E-1:
  Let Q2 := { n/[2^k] | k,n in Z with k>=0 }.

What are the units of Q2, under multiplication?  That is, what are
the numbers in Q2 which have reciprocals (in Q2, of course)?

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E-2:
  Let	L := [1+1/2+1/3+...+1/10]
and 	R := [1+1/2+1/4+1/8][1+1/3+1/9][1+1/5][1+1/7] .

Compute R-L, writing the answer -in a natural way- as a sum of
reciprocals.

  Now let
	M := Prod of [1 + 1/[p-1]] as p takes values 2,3,5,7.
Compute, as a rational number a/b, the difference M-R.

  We use that M-L is non-negative in our proof of the loglog(x)
inequality.

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E-3:
  Fix a subset S of [2 .. oo) and define its counting function T from
[1,oo) to Naturals by

	T(x) = #(S intersect [1,x]).

  Let L(x) be the sum, over all j in S with j less-equal x, of: 1/j.

  Let R(x) be T(x)/x plus this definite integral,
	\int_{1}^{x} T(u)/[u^2] du .

Show, for all  x >= 1,  that  L(x) = R(x).

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E-4:
  We showed in class, for non-negative integers a and b, that the
ratio 
	[ab]! / [[a!]^b * b!]

is necessarily an integer.

  We accomplished this by interpreting the ratio in terms of
multinomial coefficients.  Find a generalization of this --perhaps,
by using more general multinomial coefficients.


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E-11:  Given coprime positive integers A and B, show that for EACH
positive integer n there exists integer k such that
   
   A + kB  is coprime to n.

Can you give me an efficient algorithm to compute such a k?
    
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  In class I stated the following, letting  =N=  mean "cong.mod.N".  

E-12:  Generalize Wilson's thm as sketched below, and give
a complete proof.

    General Wilson's Thm: Given an odd integer N >= 3, let J(N)
    denote the number of pairs, plus/minus x, which are solns
    to x^2 =N= 1.  Then the product Prod(Phi(N)) is =N= to
    WHAT?

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