6 EXERCISES 301
53 Find a closed form for tkm,O (E)-‘(-l)kHk, when 0 6 m < n. Hint: Exercise 5.42 has the sum without the Hk factor. 54 Let n > 0. The purpose of this exercise is to show that the denominator of Bz,, is the product of all primes p such that (p-1)\(2n). a Show that S,(p) + [(p-l)\ m ] is a multiple of p, when p is prime and m > 0. b Use the result of part (a) to show that Bzn + x [(p-‘)\(2n)l p prime
= Izn is an integer.
P
Hint: It suffices to prove that, if p is any prime, the denominator of the fraction Bz,, + [(p-1)\(2n)]/p is not divisible by p. C
Prove that the denominator of Bzn is always an odd multiple of 6, and it is equal to 6 for infinitely many n.
55 Prove (6.70) as a corollary of a more general identity, by summing
and differentiating with respect to x. 56 Evaluate t k+m (;) t-1 lkkn+‘/(k- m ) in closed form as a function of the integers m and n. (The sum is over all integers k except for the value k=m.) 57 The “wraparound binomial coefficients of order 5” are defined by ((;)>
=
((nk’))
+
((,k:;mod,))’
n>O’
and ((E)) = [k=Ol. Let Q,, be the difference between the largest and smallest of these numbers in row n:
Qn = E5((L)) - o%((;)) * Find and prove a relation between Q,, and the Fibonacci numbers. 58
Find closed forms for &c Fiz” and tntO F:zn. What do you deduce about the quantity Fi,, - 4Fi - F:_,?
59 Prove that if m and n are positive integers, there exists an integer x such that F, E m (mod 3”). 60 Find all positive integers n such that either F, + 1 or F, - 1 is a prime number.
