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?
;;;; End: "e.probs.txt" ;;;;