410 Notes on Explorations
The reader may wish to explore further the intervention in the factorization
of the Fibonacci sequence 1, 1,2, 3, 5, 8,... and the related Lucas sequence
1, 3, 4, 7, 11, 18, 29, 47,.... For a reference, see Steven Schwartzman,
Factoring polynomials and Fibonacci, Math. Teacher 79 (1986)) 54-56,65.
Equations of the type x2 - &? = k come under the general heading of
Pell’s equation. Since they arise in many number theoretic problems, their
theory is covered in most elementary number theory texts. For a brief,
insightful introduction to this area and its significance, see Chapter 7 of
H. Rademacher, Higher Mathematics from an Elementary Point of View,
(Birkhauser, 1983).
E.33. For an appreciation of the role of padic numbers in the solution of
diophantine equations, see the article Diophantine equations: p-udic meth-
ods in W.J. LeVeque (ed.), Studies in number theory, Studies in Mulhe-
matics 6 (Math. Assoc. of Amer., 1969).
For other references on padic numbers, see G. Bachman, Introduction
to p-Adic Numbers and Valuation Theory, (Academic, 1964); and Kurt
Mahler, p-Adic Numbers and Their Functions, (Cambridge, 1981).
E.35. The proof of the irreducibility of Q,.,(t) requires more advanced the-
ory. See, for example, Section 53 of B.L. van der Waerden, Modern Al-
gebra, Volume I (Ungar, New York, revised edition, 1953); and Theorem
41, Chapter 12 of Jean-Pierre Tignol, Gulois’ Theory of Algebraic Equa-
tions, (Longman, 1988). In Solomon W. Golomb, Cyclotomic polynomials
and factorization theorems, Amer. Math. Monthly 85 (1978), 734-737; 88
(1981) 338-339 criteria for reducibility of Qn(tr) and factorizability over
Z of Qn(m) are discussed.
E.36. For
(P-1X9-1)
Qpq = c G?,
n=O
it is shown in Sr. Marion Beiter, The midterm coefficient of the cyclotomic
polynomial Fpq(x), Amer. Math. Monthly 71 (1964), 769-700 that
c, =
{(-l)k if n = aq + bp + k in exactly one way
0 otherwise.
Let m be the smallest value of n for which Q”(t) has coefficients other
than 0, 1, -1. By Exercise 3.5.12, it is clear that m must be odd. It can
be shown that m is not a prime power or a product of two distinct primes
(see Exploration E.35 for a reference). The smallest possibilities for m are
45,63, 75,99, 105. It turns out that m = 105 and that
Q1,,5(t) = 1 + t + t2 - t5 - t6 - 2t7 -....In fact, the coefficients of cyclotomic polynomials can be arbitrarily large.
For a discussion, consult Section 6 of R.C. Vaughan, Adventures in Arith-
metick, or: How to make good use of a Fourier transform, MathemaZical