Math 411 Spring 2009 May 4 Problems
Section 1.1, page 14, problems 1.1, 1.2 and 1.20
1.1 True or false with reasons.
(i) There is a largest integer in every nonempty set of negative integers.
(ii) There is a sequence of 13 consecutive natural numbers containing exactly 2 primes.
(iii) There are at least two primes in any sequence of 7 consecutive natural numbers.
(iv) Of all the sequences of consecutive natural numbers not containing 2 primes there is a sequence of shortest
length.
(v) 79 is a prime.
(vi) There exists a sequence of statements S(1), S(2), . . . with S(2n) true for all n≥1 and with S(2n−1) false
for every n≥1.
(vii) If mand nare natural numbers, then (mn)! = m!n!.
1.2 (i) For any n≥0 and any r6= 1, prove that
1 + r+r2+···+rn=1−rn+1
1−r.
(ii) Prove that
1 + 2 + 22+· · · + 2n= 2n+1 −1
(for all natural numbers n).
1.20 Prove that 4n+1 + 52n−1is divisible by 21 for all n≥1.