[Notations j
Let N , Z , Q , R an d C stan d respectivel y fo r th e se t o f positiv e integer s (als o
called natura l numbers) , th e se t o f integers, th e se t o f rational numbers , th e se t o f
real number s an d th e se t o f complex numbers .
Unless indicate d otherwise ,
th e letter s a , 6, c, d, z, j, k^l^m^n^r an d s stan d fo r integers ,
th e letter s p an d q stand fo r prim e numbers ,
th e letters p\, p
2
P3, PA ? Ps, P6»P7 represent th e sequence of prime num-
bers 2,3,5,7,11,13,17,.. . ,
b y twin primes, we mean a pair of prime numbers {p, q] such that q = p- f 2.
Given a n intege r n 2 , we often writ e
for it s canonical representation a s a product o f distinct prim e powers : her e the q^s
are th e prime s dividin g n writte n i n increasin g orde r an d th e exponent s a^' s ar e
positive integer s (se e Theorem 11).
[Some Classical Forms of Argument)
THEOREM
1 (Inductio n Principle) . Let S be a set of natural numbers having
the following two properties:
(i) 1 5 ,
(ii) ifk e S, then k + leS.
Then S = N.
THEOREM
2 (Pigeonhol e Principle) . If more than n objects are distributed
amongst n boxes, then one of the boxes must contain at least two objects.
THEOREM
3 (Inclusion-Exclusio n Principle) . Let A be a set containing N ele-
ments and let Pi , P2,..., P
r
be distinct properties that each element of A must
satisfy. Ifn(Pi
x
, Pi
2
,..., Pi
k
) stands for the number of elements of A having all the
properties P^ , Pi
2
,..., Pi
k
, then the number of elements of A having none of the r
properties is equal to
iV- (n(Px)+n(P
2
) + •+ n(Pr )) + (n(P
l 5
P2 )+n(P
l 5
P 3) + •+ n(P
r
_
l 5
Pr;
-(n(P
1
,P
2
,P
3
) + n(P 1,P
2
,P
4
) + -- - + n(P
r
_
2
,P
r
_ 1,P
r
))+-- -
+ ( - l ) r n ( P i , P
2
, . . . , P
r
) .
Inequalities
THEOREM
4 (Cauchy-Schwar z Inequality) . Let a±, a2 ,..., an, 61,62, ... , b
n
be
real numbers. Then
i=l ' i=l i=l
3
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