5 Further Remarks 175For the early history of Fermat’s last theorem, see Vandiver [52], Ribenboim [41]
and Kummer [28]. Further references will be given in Chapter XIII.
Arithmetical functions are discussed in Apostol [1], McCarthy [35] and
Sivaramakrishnan [48]. The term ‘Dirichlet product’ comes from the connection with
Dirichlet series, which will be considered in Chapter IX,§6. The ring of all arithmeti-
cal functions was shown to be a factorial domain by Cashwell and Everett (1959); the
result is proved in [48].
In the formf(a∧b)f(a∨b)= f(a)f(b), the concept of multiplicative func-
tion can be extended to any mapf:L→C,whereLis a lattice. M ̈obius inversion
can be extended to any locally finite partially ordered set and plays a significant role in
modern combinatorics; see Bender and Goldman [5], Rota [45] and Barnabeiet al.[4].
The early history of perfect numbers and Fermat numbers is described in
Dickson [13]. It has been proved that any odd perfect number, if such a thing exists,
must be greater than 10^300 and have at least 8 distinctprime factors. On the other
hand, if an odd perfect numberNhas at mostkdistinct prime factors, thenN< 44
k
and thus all suchNcan be found by a finite amount of computation. See te Riele [42]
and Heath-Brown [21].
The proof of the Lucas–Lehmer test for Mersenne primes follows Rosen [43] and
Bruce [8]. For the conjectured distribution of Mersenne primes, see Wagstaff [53].
The construction of regular polygons by ruler and compass is discussed in
Hadlock [19], Jacobson [24] and Morandi [36].
Much of the material in§4isalsodiscussedinMacduffee[34]andNewman[40].
Corollary 34 was proved by Hermite (1849), who later (1851) also proved
Corollary 38. Indeed the latter result is the essential content ofHermite’s normal form,
which will be encountered in Chapter VIII,§2.
It is clear that Corollary 34 remains valid if the underlying ringZis replaced by
any principal ideal domain. There have recently been some noteworthy extensions to
more general rings. It may be asked, for an arbitrary commutative ringRand any
a 1 ,...,an∈R, does there exist an invertiblen×nmatrixUwith entries fromRwhich
hasa 1 ,...,anas its first row? It is obviously necessary that there existx 1 ,...,xn∈R
such that
a 1 x 1 +···+anxn= 1 ,i.e. that the ideal generated bya 1 ,...,anbe the whole ringR.Ifn=2, this necessary
condition is also sufficient, by the same observation as when invertibility of matrices
was first considered forR=Z.However,ifn>2 there exist even factorial domains
Rfor which the condition is not sufficient. In 1976 Quillen and Suslin independently
proved the twenty-year-old conjecture that itissufficient ifR=K[t 1 ,...,td]isthe
ring of polynomials in finitely many indeterminates with coefficients from an arbitrary
fieldK.
By pursuing an analogy between projective modules in algebra and vector bundles
in topology, Serre (1955) had been led to conjecture that, forR=K[t 1 ,...,td], if an
R-module has a finite basis and is the direct sum of two submodules, then each of these
submodules has a finite basis. Seshadri (1958) proved the conjecture ford=2andin
the same year Serre showed that, for arbitraryd, it would follow from the result which
Quillen and Suslin subsequently proved.