Concrete mathematics : a foundation for computer science

What’s 114 in
radix 11 ?
for all c > 0.
A ANSWERS TO EXERCISES 511
4.70 This holds if and only if ~2 (n) = 1/3(n), according to exercise 24. The
methods of [78] may help to crack this conjecture.
4.71 When k = 3 the smallest solution is n = 4700063497 = 19.47.5263229;
no other solutions are known in this case.
4.72 This is known to be true for infinitely many values of a, including -1
(of course) and 0 (not so obviously). Lehmer [199] has a famous conjecture
that cp(n)\(n - 1) if and only if n is prime.
4.73 This is known to be equivalent to the Riemann hypothesis (that all
zeros of the complex zeta function with real part between 0 and 1 have real
part equal to l/2).
4.74 Experimental evidence suggests that there are about p( 1 - 1 /e) distinct
values, just as if the factorials were randomly distributed modulo p.
5.1 (11): = (14641),, in any number system of radix r 3 7, because of the
binomial theorem.
5.2 The ratio (Karl)/ = (n- k)/(k+ 1) is < 1 when k 3 Ln/2J and 3 1
when k < [n/2], so the maximum occurs when k = [n/2] and k = [n/2].
5.3 Expand into factorials. Both products are equal to f(n)/f(n - k)f(k),
where f(n) = (n+ l)!n! (n- l)!.
5.4 (-,‘) = (-l)k(k+;P’) = (-l)‘(;) = (-l)k[k>O].
If 0 < k < p, there’s a p in the numerator of (E) with nothing to cancel
t% the denominator. Since (E) = (“i’) + (:I;), we must have (“i’) = (-l)k
(mod p), for 0 < k c p.
5.6 The crucial step (after second down) should be
The original derivation forgot to include this extra term, which is [n = 01.