Number Theory: An Introduction to Mathematics

7 Further Remarks 581

Elkies (1988) used the arithmetic of ellipticcurves to find infinitely many counterex-
amples forn=4, the simplest being

958004 + 2175194 + 4145604 = 4224814.

A prize has been offered by Beal (1997) for a proof or disproof of his conjecture
that the equation

xl+ym=zn

has no solution in coprime positive integersx,y,zifl,m,nare integers>2. (The
exponent 2 must be excluded since, for example, 2^5 + 72 = 34 and 2^7 + 173 = 712 .)
Will Beal’s conjecture turn out to be like Fermat’s or like Euler’s?

7 FurtherRemarks

For sums of squares, see Grosswald [31], Rademacher [46], and Volume II, Chapter IX
of Dickson [23]. A recent contribution is Milne [42].
A general reference for the theory of partitions is Andrews [2]. Proposition 4 is
often referred to asEuler’s pentagonal number theorem,sincem( 3 m− 1 )/ 2 (m> 1 )
represents the number of dots needed to construct successively larger and larger pen-
tagons. A direct proof of the combinatorialinterpretation of Proposition 4 was given
by Franklin (1881). It is reproduced in Andrews [2] and in van Lint and Wilson [41].
The replacement of proofs using generating functions by purely combinatorial proofs
has become quite an industry; see, for example, Bressoud and Zeilberger [13], [14].
Besides theq-difference equations used in the proof of Proposition 5, there are also
q -integrals:
∫a

0

f(x)dqx:=

∑

n≥ 0

f(aqn)(aqn−aqn+^1 ).

Theq-binomial coefficients(mentioned in§2 of Chapter II)
[
n
m

]

=

[

n

m

]

q

:=(q)n/(q)m(q)n−m ( 0 ≤m<n),

where(a) 0 =1and

(a)n=( 1 −a)( 1 −aq)···( 1 −aqn−^1 )(n≥ 1 ),

have recurrence properties similar to those of ordinary binomial coefficients:
[
n
m

]

=

[

n− 1

m− 1

]

+qm

[

n− 1

m

]

=

[

n− 1

m

]

+qn−m

[

n− 1

m− 1

]

( 0 <m<n).

Theq -hypergeometric series
∑

n≥ 0

(a)n(b)nxn/(c)n(q)n