the same
a
large
and
on
the
other
extreme
of the
presents in twenty chapters
seems appropriate
an
and for the remaining
the nature
Fermat
discovered
name
and
which
itself
. Euclid
(about A. D.
it
cor-
p.
Very
many
fifth
perfect
number.
127, 257;
2, 3, 5, 7,
number
must
Dickson
(p.
30).
whose
divisors
equals
a
were actively
^
and
amicable
the sum
n
is
reserved
for Chapter II, in which is presented the history of
Fermat's two
problems to
had
a
by
p,
then
x^~^
The
relatively
. .
zation of Wilson's theorem: if
P denotes the product
one
of
generalizations
in which
Fermat's
theorem :
if
a^
Any integral
properties of the quotient
by Cayley in
ji\i
column any
number such that
was
first
proved
of the
compact
language,
differ-
ence
congruent
modulo
m
that n belongs
1773.
In
called the
For
a
prime or
power of
if
has
not
only
led
to
an
important
extension
to wide
modulo
p,
f{x)
logical
basis
of
the
theory.
expression
of n,
odd
divisors,
or
sth
in
the
pp.
330-1;
theorems
on
odd
11 or 7 to
topic
The frequent need of the factors
of numbers
third
Numbers from 1 to
account
Absolute
accuracy
ways, or by use
 
primes
each
of
38 and
and
of
certain
trinomials.
In
Chapter
XVII
are
treated
questions
on
the
divisors
Euchd gave a
along
with
generalizations
and
related
the
first
of
several
immediately
successive
citations
of
yet
pages
experts in
the
initial
page
Maillet,
L.
S.
and
Kempner,
To these
eleven men
due
the
gratitude
Finally, this laborious
project would doubtless
approved
and
and
Wallis
51
III.
10
181
By the
a
number
are
meant
divisors is called perfect
the
types. The
between
arranged
with
fitting
order;
6,
28,
496,
one
perfect
number
and the
number,
God
perfection
by
the
number
it; likewise for 10. But the sum of the aliquot
parts
of
12
exceeds
it.
are
6,
28,
496,
end alternately in 6
Isidorus
6
is
a
perfect
which
sprung
the
entire
imperfect than
^lamblichus Chalcidensis
explicatio in duos
 Arithmetica boetij, Augsburg,
prime;
<0.
Hrotsvitha,^^
a
mentioned
the
even.
Leonardo
Pisano,
or
Fibonacci,
Geschichte
Math.
Wiss.,
1880,
for finding
wahrend
der
Spanish
Arabischen
Periode,
Wien,
1873.
*^In
hoc
Hbros
arithmeticos
diui
Seuerini
Boetij.
part
1461,
(J.
33')
factor
odd number equals
expressly
committed
rule.
1455,
tl537)
codices
nuna.
11001-15028
1495,
7th
unnumbered
page.
Arithmetica
SpeculativaThome
always
be
found
alternately.
Carolus
^
the digits
and remarked
the
implication
in
6
and
8.
Regius^^
defined
a
2047 are
the perfect
Perfectis
Liber
Nouamente
stampato.
Venice,
1526
number
Cardan^^
(1501-1576)
numbers,
6
the multiples
of
5,
etc.
WilUchius^'^
(tl552)
6
e
augmentee
par
1'
Auteur,
Lion,
1554
under
6-10^
are
6,
28,
(p.
29)
of
his
Arith-
meticae
or fewer
end alternately in
number
6.
He
verified
(1585,
pars
k
Lazaro
Schonero
recog-
niti
of
a
perfect
number
and others, that all
(pp.
12-17)
by
actually
trying
composite.
He
proved
(p.
8)
alternately
in
6
and
8,
powers of
factor
2 —
are
given
by
perfect
number
is
triangular.
16C3. According
,
property
that
Casper
Ens.^^
Erycius
Puteanus^^
the
union
even.
Puteanus
in order
remaining
there is no perfect
Mathematicarum,
quas
geometry
and
one
for
arithmetic.
 Thaumaturgus
31
if
number is of Euclid's
(?),
considerable
discovery
2 ^^-
has
Brassinne,
(4),
3,
1853,
149-150.
•°F.
Marini
and Frenicle, that
by
W.
10,
are
thus
=
would find 11
remember
assigned
so
what-
ever
exponent.
Thus
496
256.
*T.
Marini
Mersenni
with
p.
63;
Cap.
than 11 and
hundred
which
to Car-
135-140):
those
with
between
10
and
100,
8.
Andrea
Tacquet^^
given
as
if such
378,
Heildelbergae,
that
2^^
numbers
when the odd
Profitable, London,
recensione Henrici Siveri,
John Hiir^
when d
conclusion
y
final factor shall
is not
number not in Euclid's set. He stated that, if pA
is
factor
of
A,
then
and
ed.
2,
1801,
546-8).
 Recreations
&
Zahl).
223,
1,
2,
3,
numbers.
[For
later
proofs
by
Winsheim,®^
were listed
successive powers of
those
for
n
[Chap,
i
perhaps
number since it was
Hansch^ that
^
and none
math,
to
pri-
vately
in
L.
2 =»=
lowest
terms.
sum
of
not less
L. Euler^^
must be
of the
.the
odd number,
and the
by Lionnet,^^^
cited [Paciuolo^®]
perfect
numbers
have
been
For a
notation
and
less
remarkable
than
novel,
London,
1823,
131.
1831.
Numbers.
21
the
impossibility
Desboves^^'].
belongs
modulo
p,
For 34
2^^
But
the
minunum
3,
Stutt-
gart,
1856,
61
pp.
Described
by
Kummer,
Jour,
fur
Math.,
had been given
sum
of
the
primes.
E. Catalan^^^ remarked that, if we admit the last statement, and
note
that
2^
base 2.
where are given
numbers.
He
stated
(p.
162)
that
he
had
the
plan
with
=
4m
follows
from
Euler's
criterion
that
2^^ ^^''^
491.
Lucas^^^
elsewhere
stated
(a
left
are
zero.
The
[Chap,
i
H.
LeLasseur
<30000 of 2 -l for
the
127, 137,
(of
difference
<y-\-l.
a
LeLasseur^^^ on the 24 prime
values
and
for
a
par
M.
Jules
Carvallo,
Paris,
1883,
32
pp.
 <Mathesis,
6,
1886,
147.
are
primes
such
that
odd perfect
LeLasseur.^^^
The
primality
considered
composite
by
3^^1(mod
is
stated
without
/S
and
F
of
Euclid's
p<r.
Jules
a-+i_l
proved
that,
sum
divisors
of
a
perfect
number
equals
2.
CI. Servais^^^ showed
.
.
.
whence 2(a — l)<a+n
—
number a' 6'*. For the case
of three prime
manner. He
proved that
prime
factors
number is not divisible
form
7k='=
1
any
one
13,
etc.
E.
Lucas^
by
Euler,^^
2
2''
two
papers.
were
di
of
[Catalan
6/cH-5.
F.
J.
Studnicka^^^
called
Ep
Ep is E/~'^.
p
prime
by
a
.p/
it,
so
=2^ *,
2*,
has at least four distinct prime factors and exceeds 2000000.
A.
respectively
5136
divisors of
where
p
is
forms f^au^
Frenicle.^'
What
Mersenne^
actually
etc.,
in
the
have
a
under one
composite
(Lucas,^^°
Cole^^^),
while
Mqi
known factors
+
v-vi.
11,
1908,
36-8,
53,
75-6
(Cipolla).
P.
as an
a
letter
in
which
he
gave
. . .
the fourth
than
the
formula,
but
finding
them
find,
for
example,
a
discovery
aliquot
parts.
V.
If
n
P4<3^
P^^^K
P5 (com-
his correspondents,
without cita-
Ps^\
and
also
P4^«)
=45532800
stated that
it is
34 P4,
far.
In
1652,
Ozanam's'^
Recreations,
I,
p.
35,
edition,
I,
p.
39.
Fermat.
The
to
Mat.
e
Fis.,
12,
1879,
714.
V\'V2Vzy
to be
suitably determined.
the
only
one
given
q.
He
found
in
1902
(but
'20Bull. Bibl.
'^Questions
has
only
n
distinct
prime
factors.
Carmichael's^^^
whose
divisors <
divisors of
the other.
contrary, amicable num-
the
for amicable numbers was se invicem amantes. In the article
in
congeneres,
A. D.):
number 220
who have concerned
and presents
to the
moon an
sum
of
whose
friendship
is
sought.
The
of
''giving
any
one
the
smaller
21,
I,
1868,
178-9.
 ^'Manuscript
Magriti;
=
y,
z
odd
primes];
then
2yz,
or
^^
derived value.
de
Fermat,
4,
=
pairs
[including
two
false
which
the
common
factor
a
is
relatively
jm
for
expressing
6^
=
=
resulting
cases,
p,
q,
ars, where
p, q,
r, s
Since
jP'j2
=
member as
a prod-
the
by
ff=hg
andp+l
is
to
—
M+bff
N+bff
zap, zbq, where a and 6 are given numbers, and
q
6,
p,
q.
Set
JoM
6
relativelyprime. Since(p-fl)j a
a
common
factor
the cases
ible by
to 22Z
k
as
in
' ^Bibliotheca
Math.,
(3),
14,
1915,
351-4.
in
(2),
(3i),
(63);
jS
twice
in
(3i);
7
in
(2).
Moreover,
list,
a
fact
noted
on
p.
these
three
pairs
occur
the
list
his posthumous
tract®^ on
72J
3i,
p.
59,
note
320,
work.^^®
while PQ
the
square
of
the
form
s^
num-
bers
=
vol.
2,
pt.
2,
1807,
34-39.
<ni
of
?ii,
etc.,
35 the numbers
=
amicable
(a),
(^3)
is a
are
amicable
if s^(n)=n and raised the question of the existence
of
numbers
n
.
known
2*-12959-50231,
These
to him.
of
chain
n,
said
to
be
of
chain of period
3, 4, 5,
Generalizations
and 35 have the same sum of
ahquot
true of the
51,
91;
95,
119;
69,
such that the product of
n
of
them
shall
293-3370,
5-
Plato) has 60
the
number
long problem
a^bc,
parts.
van
rule that
iPractica
Arith.
square.
Jean
Prestet^
n+1
being
not
2*3^5^
of a '?) .
E.
Lionnet^^
the product of
2^3
ha\'ing
the
Divisors
of the
and
factor
in
otherwise
by
Wolff.
L.
Euler^^
is just
and
of divisors:
a
square.
Problems
for example,
posed
= y^:
by
31
is
993,
which
equals
the
of
the
 A of
the base
x, he in effect put X={x^-\-a)^, 0<a<x. Then
a^
and
F
=
problem
(i)
x
 Nouv.
Ann.
Math.,
(3),
2,
1883,
a{SnV): a{2'S')
form
x
biquadrate.
E.
Lionnet ^
ones
are
p^
and
pq,
where
p
and
q
P.
de
Fermat^
while
investigating
that he
in
those
Fermat's
pyramidal numbers, that
(mod
3),
and
similarly
induction)
by the
as
above.'*
That
the
binomial
coefficients
in
(1 +
1)^
k. He remarked that Carr^
had
He was
(a+2)''-(a+2),. .
a
p.
it
avoids
not
is
12, 1853,
49;
Werke,
a ,
where
.,
of
.
.
.,
a^k.
those of the
It
 a,
Comparing
with
the
original
equation
multiplied
by
. . .
for the
-(^3^)(p-3)^-^+.
p,
and
applying
prime if
which
reduces
to
(1)
—1,
then
by the first
=
method
de I'lnstitut
1776,
16-23.
Bossut's
Algdbre,
theorem, V,
theorem and
noted that
he
a
two
in
any
pair
is
two
(n
a prime
1
1
and
1
1
residues
Toulouse,
3,
1788
(read
Dec.
4,
1783),
p.
91.
two
of
a
xn+1.
be
Gauss^*^
the
theorem:
by a proof
Euler's^° first proof.
by
p
(thus
generahzing
Euler's-^
The case a
quoted
by
Gnmert,
Archiv
Ivory
by
Ivory
modulo n.
p,
p'
by
pp' . .
we see
J.
A.
2k,.
p.
1095)
we
square
set
(6),
1,
[Chap, hi
p,
a,
/3,
p),
quadratic non-residue of
First,
let
A
Raising the above a*=—
1 to the power
holds
moduli
g ,
r^
1 is a
the case
of Ivory
pth
x^=l
2 ^.
For
p
Nantes,
1836.
 London
and
Edinb.
Phil.
Mag.,
11,
1837,
456.
 M6m.
divers
savants
ac.
sc.
Institut
de
France
(math.),
5,
1838,
19.
 Phil.
Mag.,
prime
to
=
+
is
two
factors
u,
v,
whose
g.
c.
d.
the num-
distinct
odd
primes
is
odd
if
s
A. Cauchy^^
products of the roots
then
AoSi
After omitting the numbers
obtain
of
these
by h are prime to A
and
hence
included
11, 13,
of
the
l
Wilson
of
number n of pairs of roots ±x of x^=r
(mod z)
he stated
=^x of
x^^l
(mod s)
two factors P,
and let
N
and
initial
polygon
we
take
the
remainder
(mod N),
—
p
or
If
m at
1,
2,
the
 Periodico
ed.
2,
1854,
324.
^«Proc.
factor
p
G.
L.
(§38)
C.
period a2.
.
^
p
positions.
Hence
p
divides
(p
Cf.
are not
q^
divide
100. .
.by
divides
R'^'-^
such
has
roots
if
of m
—
sum
of
the
products
of
1,
we get
the argument
a
proof.
E.
a
prime,
n\
is
divisible
p
at
a
time,
a
those containing elements
each
combination
--iV-tO^-
 'BuU.
Soc.
Math.
France,
11,
1882-3,
If ^(n)
the greatest exponent of the
prime factors
of n.
 ^].
two ways
Each
prime, this
on subgroups
a group,
are
(i
+
a
prime,
and Fermat's
(mod
g ).
to obtain
theorem.
L.
E.
p^(p
to
any
modulus
of
Km
and
prime
vaUd for
1212
by
phisms
theorem.
K.
2^~^
/3,
7,
In
.
.)^
=
</)-function is the
multiple
of
n.
A
practical
method
of
result
by
Lucas.^^°
J.
A.
Donaldson^^°
by
different
kinds
n
where
D
ranges
theorem
odd
integers
<2^.
Then
rj,
be
=1
(mod
2'' ''^).
Next,
79-84.
 i/bwf.,
78-79.
 ^Tagbl.
/S„ of the products taken n at a time of
1, 2,
it
follows
by
by a
and taking
of Wilson's
18,
[Chap, hi
is prime to
and
Wilson's
theorems.
E.
of
irre-
ducible
congruences
modulo
this
relation
read-
ily
leads
F (a,
N). [See
N)
is
divisible
by
N.
For
. He
obtained
(1)
lengthy method
completed for
special cases.
relations
(2)
np ) ^F{a''\ n) -F{a'' \ n), F^a, p')=a^'-a'' \
where
a
by
the
prime
p.
-f-
the
index
the
in the
numerator and
roots belonging
divisible
by
n.
MacMahon's^^^
paper
contains
N)
result that F{p,
p
like
conclusion
was
 Casopis,
Prag,
that
similar
to
to
ABC
if, when d
k).
The
analogous
extension
of
same
ible
For
general P{x) is proved to
be
iMx)Mx),
Mx)
the
k=l
k=l
arbitrary
poly-
nomials.
Xi'' *-^
X
p* ^
XiP
Xi
m
ai,
{x-ai)
. .(a:-a,)
(1)
• • •
. .
 
Borel and
powers
of
Gi,
02,...,
Op-it
have incongruent
the respective tenns of
02)+02,
• •
•'
'
  ,=;
P..=
:^a;^-'/Dj,
Dj
p).
Let
p
be
a
prime.
n ). Next, let
—
by
p
—
1,
ai,
compare
coefficients.
Hence
Taking
a prime. If n
coefficient
( ) was
shown
to
be
not
divisible
by
then d'
(since
divisible
—
\,
—
the
period
for
1/^
is
2 =*=
1
is
or
is
not
divisible
a ~^
A^
is
a
product
decreased
by
unity
a
prime
to
n,
quadratic non-residue
*F.
Thaarup^^'
a
divisor
of
10^ ^
divides (2*+l)/3
necessary.
J,
H.
for
pg
that
they
Fermat's
theorem.
M.
Cipolla^^^
stated
if
for which a^~^
find
P,
any odd factor of (a^ — l)/(a^
—
residue
of
q,
=
more than
N
holds for every e
a
prime
if a: ~^=l (mod n) for all integers x prime to n
and noted that this
is false when n is the square or higher power
of
and only if P
by
X(P). The latter condition requires that, if P is composite, it be a product
of three or
146-8.
 'Math.
Quest.
Educat.
Times,
two k's.
by
Lagrange,
If
and
s is divisible by
(mod a ),
(mod a^).
If a,h,. .
positive
integers
<p
1,
[Chap,
hi
J.
and
others.^^^
s^
is
divisible
by
p
the terms
J. J.
numbers
chosen
from
1,.
expansion
of
47,
1887,
145-6;
«
contain
k
is
Wolstenholme.^^^
N.
M.
by
[Lagrange^^],
J. W.
L. Glaisher^^^
stated theorems
on the
r
3,
>1,
and
if
w
n+1
[Chap. Ill
{2n)\Z\n/r.
^„_2(1^...,
\m-l\^),
^2n.-4(l,...,2m-l)
=
divisors
of
Ar
and
Sr.
For
r
odd,
prime
>3,
by
m.^
p^
and
p^
and
proved
that
(72,
o'i,
t
multiple
of
p
this
not
divisible
is
successive odd integers,
i
form
If
A;
by
p.
Let
p
least prime
if
p
p
Take
M
and
of
the
J{\)
r%i,
g^=eS
residue
modulo
r
p'),
487.
induction that, if
Xp=(a^'-a '
.
.,
(1 +
1)^,
op_2
1 /
1
the work
5=1
solutions
<p
oi
ixv
p
N:
A'
»-l
N
of
incongruent powers of a,
Sylvester's theorem,
T T
—
A.
Friedmann
and
J.
p^
for
a
prime
p
Raay^^
=
+
...
p),
summed for all sets of solutions of s^=f^+l (mod
p).
Finally,
*H. Brocard^^
each
prime
least
residue
32aSphinx-Oedipe,
7,
1912,
4-6.
It
is
stated
that
G.
(modp2),
base 2.
investigated
the
a
have
the
same
remainder.
If
p
is
a
prime,
(^(p )
prime to A.
Then the integers
<^(J5) integers
A
=
p''{M—ii)
integers ^Mp and
the
Mp^~^
multiples
^Mp^
of
p,
exclude
p^'V
=
There
that
of N).
S0(d)
integers
^N,
proved
of the numbers
by
both
p
and
q;
Mp
and
if
0,X,ju,
6,.
divisible by
=
the dis-
the divisors
p
chosen
from
rii,
between
kaA
and
{k-\-\)aA
A:
pi,
p
if
d
is a
product cd.
. .kl oi
factor
.
.
,
with
the
factor
I.
others have one of
But
.
.,
results,
(2)
is
easily
deduced
[cf.
prime
to
N
of
degree
a
divisor
of
N.
values
1,
Xi
is
prime
to
ei
are divisible
divisible
by
no
one
of
the
dn,
\1/{E)
j&rst
that
(p.
74)
that,
when
which leads to
Sum
/3i
by
i of the
every
integer
first theorem
for n
Nip-l){q-l)...{k-l)
s=lLsJ
is
of
p
is
p:p
set
S
of
with
the
in
S
symbolically.
Hence
(ha)S
{.
\l-{c)\
With Silva, let
in S{ab), . .
m,
we
get
(to
be
explained in the chapter on that subject), which for the special
case of
</)(n) becomes
Hence
<^(15)
sets
1,
2,
3,*
4,
5;
6,*
multiples
5
odd integers.
in the
expansion of
5i
5;
then
0-0-
the
given.
numbers
which
which
a
it
gives
a
4>{z)
(2)
instead
of
(3)
formula
since [n/r]
according as n is or is not divisible by r,
v^^J^
w(n+l)
The constant
2{*(i')(l-+2-+...
 London
G.
L.
proving
(3),
obtaining
N,
there
are
exactly
</)(d)
integers
^N
and
wrote
continuous variation
of m.
0(1)+
that
0(m)
primes,
<^(m)
if ^
A^=A^_i0(m),
/(§)
=
(i>k{'m)
column is/(5),
Niedere
Zahlentheorie,
I,
1902,
97-8.]
repeated
his*^
''generalization.
.
.
TaUk)=a'-'b^-'c'-\
Then,
iV^ 100.
in the
n^
Sylvester^^
stated
the
is equivalent
Sylvester, which
accords with
proof
of
<^(n) is Qn/w^
prime
factors.
or
0,
 M6m.
Soc.
R.
Sc.
de
LiSge,
(2),
10,
1883,
No.
6,
74.
 Atti
Reale
Accad.
Lincei,
(4),
1,
1884-5,
709-711.
for 2
the
case
in
which
In particular,
where
j
+ 1,
equals
the
number
of
prime
If F{n) =S/(d),
S7,(d)
integers
which
with
n
have
the
g.
sum
Ces^ro^^ factored determinants of the tj^e in his
paper,^^ the
finding
of
known fact that
G/tt ). On
n.
iV
If
we
81,
426;
=
all its subgroups and the simple groups
whose
direct
are
<f){n)
[Prouhet^sj.
. .
.
3.
C(n)
Fermat's
theorem.
A.
E. Jablonski'^^ considered rectilinear
permutations of indices a,
=
permutations of indices a'n, .
If
permutations
of
indices
a, . .
of
the
given
positive
integers
rii,
n's. The
formulas
formula
(see
Ch.
XIX).
J.
E.
Steggair^
prime,
a/3
is
a
primitive
a6th
root
n
K. Th.
where
d
ranges
over
/3
read
r[n]=0(l)+.
star
polygons
to the
di
(/,
m
x
x~^
primes
12, 18, 24,
ha\dng no square
only
prime
factors.
is odd
.,
=
>3,
r9^2q-\-l.
whose
order
in the
F^(l)
F^(2)
their
product.
if aj is an nth root of unity, and if
c?,i,
.
numbers of the
cyclic
group
p
is
of
integers
$(aS) of
[Chap,
v
not.
Hence
that, if x ranges over all the positive integers for
wliich [m/x]
is odd,
values of the
fractions m (P)
]Miller^^^ defined the order of a modulo m to be
the
least
positive
p
relatively
integers
and
d.
Hence
7n
rpi,.
and
—
\).
factor of 7i,
remark of
where
x,
y
are
if
x,
y
m.
Also
S
(f>kimn)=(l)k{ni)(t)k{n),
f
•'
to n or only over
such
is n.
=
..
like that
.,
be
of the primes
1)
in arith-
deleting
a,
/3
by
A^ —
j8,
where
but not if
a/N,
^/N,
d
A^,
and
u
is odd, then
the
+
arithmetic mean
of the
v
p,
<l)m{N)
all divisors of n which have some definite property P,
while
pth
When
this
result
becomes
Thacker's.^^
N.
Nielsen^^^
(Math.),
relatively
prime,
6
(modm).
For
sets
is
*/c(n).
himself
<b, etc.; there
—
linear
(f>k(n),
sets
of
k
integers
^n,
while
{n/piY
A.
Blind^^^
kind
is
by
we
do
not
Story-°-
defined
 the
kih. totient of n to the condition k to be the num-
ber
sets
of
k
numbers
the
con-
dition
X
is
(if
different
permutations
which
J. W.
while
pi,
P2>
generalization
of order n in which the elements of the last
coluimi are
in the (s
1 or
polygons
of
n
with
(12)
and
Formula
(14)
by
replacing
function
/^(n).
R.
D.
z'l,
mi,
by Hacks^^ of Ch.
them
and
[x].
derivatives
numbers
n
[n,
gj '
of
G
is of period 5 if and only if 5 is the least
positive
value
5 and
N
=
multiple
totient
of
Jm(ky)=JM
equations
by
replacing
x,
??2,
n,
^^i^,
n>2.
A set of m integers (not necessarily positive) having no common
divisor
a
totient
point.
Xi
the figure
=
plane, in
which every
incongruent sets of
to a double
their
lowest
terms,
denominators
it
consecu-
tive
fractions
of
a
Farey
=
b+b
two fractions
e.,
have
the
sum
unity),
the
same
*A.
Brocot^^''
M. d'Ocagne^^'^
p
by
adding
2''<n^2'+\
be
relatively
rational
divisible
a
period
10 *-
1,.
prime
it
divides
for 1/D is
multiply l/D
divisors of
10*±1,
and obtained
=
latter's^
divisors
of
10''
occurs, then
10^+1
Henry Clarke^^
discussed the
(1773),
273-317.
table for converting
m
computed^^
the
period
for
a/b,
the period for a/17
cyclically
from
that
for
1/17.
Goodwyn^^
half
periods
is
9,
theorie,
1887,
153-6.
I'Jour.
Nat.
Phil.
Chem.
Arts
(ed.,
nombres
du
a
volume,
the
^
the
one
numbers
of a series of concise and useful tables of all
the complete decimal
divisor, by
all
integers from 1 to 1024. To which is now added a tabular series of complete decimal
quotients for all
terms, neither
the numerator
nor the
by J.
Journal, London,
47, 1816,
fractions
of
is
any
not
divisible
by
p,
S
of the
35,
quotients
c.
have 2n
=
Similar results
hold when the period of mn digits is divided into n
parts
of
m
digits
each.
002481389578163771712158808933
for
1/403,
complementary
(10^^ —
as a/x-\-
ai/x^-l-
^^Polytechnisches
Journal
(ed.,
'^'De
quelques
a
multiple
a
a
d'
have
no
or
5,
know
the
1/p,
1/p^, .
for
same length
theorem,
the sum
p
is
a
the period
The period
if and
k
the
first
exponent
giving
a
»»Nouv.
Ann.
Math.,
1,
1842,
464-5,
467-9.
*nhid.,
2,
1843,
80-89.
 /bid.,
5,
1846,
661.
^IhU.,
3,
1844,
applied only to the
that for
given
primality
stated
(pp.
294-5)
for A/P^ is
—
by 5.
a
by
5,
and
for
n
odd
if
a
has
3^~^
digits
each
prime
p<
150.
period
28,
30,
32,
36,
42;
those
of
analytic
factors.
with
71(10™
—
half periods
dividing
10
/8
[Bernoulli^].
Let
p
have
a
A-
Let
ri_iB
nat.
de
Bordeaux,
3,
1864,
245.
G.
Salmon^^
remarked
that
Shanks^- gave primes
same
any
A
stated
having an even
the half periods
has all its
of x is ^a,
we see that Z/N
the first
modulus
He
asked
if
6
the
sum
G.
Bellavitis^*
noted
by
5;
for
1/499,
periods form
For
q
the period
4>{q)
cyclic
order;
the
counting from the entries in Good-
wyn's^^
character of this
for the lengths of periods.
The
for
which
the
property
1/p,
1/pi,
l/p2,
are
s,
Si,
S2,
Si,
N
from
the
exponent
of
the
same
prime
factor
to
it,
we
primitive root of h',
relatively prime
to P,
fractions.
There
is
here
first
published
(pp.
36-41)
Reuschle.'*
Cunningham^^^
of the concept of
A{N^
is an
math.,
5,
1898,
57-8;
10,
1903,
91-3.
N.
of
10^^
,
the period
e.,
the
exponent
to
for
(to m
binomial
congruences
[see
[Chap.vi
on
higher
residues
from the latter
Ci
^^
Reuschle's'*'^
be a
modulo
p.
on mixed fractions.
theorem that,
28,
1910,
22.
for
the
p.
178).
theory.
W.
H.
Jackson^^^
noted
that,
if
be
written
as
shown how to find
1/D. Again/^^
for z/p.
digit of P^. In
the tens digit
.
only if n
is a divisor
(arts.
x
(1),
p.
If
^•^^1,
^-^-^Kmodp),
/3
j8'
to
C.
roots of
d
modulo
p.
Let
ypid)
p,
(mod
p-1).
Gauss
(arts.
69-72)
convenient
base.
primitive root of
dues of the successive powers of
2);
2,
and
p,
nary into
decimal fractions
of anyodd
are
general, the algebraic expres-
unity
Paris,
ed.
3,
1804,
of
p
If
p
roots
are divisors of I, and if
we
replace
form,
we
get
a
the
roots
the product of all the numbers belonging
to
an
exponent
d
is
=
(
—
the expo-
nent n,
occurs in
a primitive root of
powers
(p
p
is
1,
the
negative
of
p
natural
comes the
for a modulus
Given
any
numbers.
(See
Jacobins
and
Tchebychef
.3^)
prime
and
power
given indices and the
the table
and 73 of
 Anfangsgrlinde
to
the
least
/
we have
chosen
so
proceed
similarly.
used for the
Jacobi proved
a
primitive
exponent
e
modulo
p,
and
exponent
ee'/8
modulo
pp'.
He
discussed
at
i
base m
base m
nj,
112,
ii,
io
are
factor in
is
integers <n
correspond
to
a
/i,
/2,
=
maximum indicator
<^(p )
is
the same
the existence of
4>{n) integers belonging to the exponent n, sl divisor of
p
own^
exponent t
numbers belonging
to
give
a
uniform
method
belonging
to
to
2 *
modulo
2
are
a prime
. There is given
errata
in
Jacobi's^^
Canon
(p.
222)
n
modulo
p,
a
prime.
announced
the
theorem
(p.
283)
that
a
prime
factor of
roots of
conclusion
as
root of 2a
a
prime
10,
2,
of
p;
(b)
1000<
p<
5000
and
a
e Sc.
thdorie
general
results
are
erroneous
which
1,
they
would
p).
Hence
and
x\
for
table showing the
=
for each N) of
^
the
moduU
2
and
5
< 10160.
V.
Bouniakowsky^^
prime
p
p
and
{p
and
(p
following
theorem
/
which
root a'6 , where
.,
the prime
but
follows;
two series of positive
exponent
divides a *
the
irreducible
fraction
N/B.
L.
Sancery^^
positive
integer.
If
A
belongs
exponent
6
modulo
p
p
[Barillari^° ].
Hence
A
the
exponent
6,
exponent
3
unless
p
Fritz
Hofmann^^
used
rotations
of
a prime (Gauss^).
of
the
primitive
roots
of
a
exceeding
prime
primes
number of primitive roots.
indicator of n, and noted that it is a
divisor
which a is
powders
whose
in
Serret's
Algebre,
2,
No.
318,
2,
1891,
478.
respect to which
and
qi
the
=
(mod
qi)
for
the prime
except in the following
7.
As
a
corollary,
every
of
a
2q-\rX
is
a
G.
F.
Bennett^°
proved
(pp.
196-7)
the
first
theorem
.
.)
to an
say
1,
1,
—1,
—1.
Either
evaluated
exponent
2 ^
modulo
2
the
form
quasi
primitive
roots
of
8A:+1 or
two
numbers
2 ^
two numbers
an index by one without
an
belonging to the exponent
an
numbers
belonging
the
k divides
and
Gauss'
lOyc+9,
lO/c+7.
between
He noted
prime
to
N,
certain
higher
values
a
given
value
The
tests
are
the
simple
quadratic
or
cubic
residue
of
p.
If
prime
To every
the roots
of the cubes of the numbers which remain after excluding
the
residues
primitive roots [evident].
^^
p
<
indices
a'''+l=0
divisible
by
6],
Soc.
Kasan,
(2),
indicator
is
manuscript to
modulus
Epstein's
formula
requires
modification
when
m
2dodi
(f-^)\.
British Assoc. Report,
are
of primitive
A. N.
Korkine ^ gave
g^p~'^^^^
for k
of Jacobi^^
=
belongs
indices
for
having
a
given
index.
putation.
If
X
belongs
a
q.
The
more than one if
=
1,
=f1.
When
q
is
even, x is said to be of the first or second
kind according
kind,
we
get
a
g
is
a
possible
even
p<
for
which
y
p'''<
10000
for
z
<m
and
prime
to
belongs
p.
On
exponents
Binomial
Congruences.
Bhdscara
Achd,rya^*^
(1150
with corresponding conditions for impos-
sibihty
(§206,
p.
265).
For
y^
odd, and that the
a}-\-hx=
and proceed
69+
we reach
5^=3
is
a
residue
of
an
ax-\-b
b,
square
p^)
by
setting
x
be a
by
2'',
must
be
divisible
by
2 -''.
If
f
±p
must
be
by
fg
ii
x=
jif^
p
is
a
co
roots
of
(1)
and
(2),
by
use
of
(3).
Set
n
an
integer
(mod
p).
Hence
by
we reduce
particular, let n
=
p,
added
—
by
p.
=
best
If we
is
divisible
by
p .
[Cf.
=
of N; the
a. If
odd prime
any composite
modulus to
the case
[Chap.
VII
the
proposed
P. S.
terms
binomials
equality
implies
that
a
p
roots
p
roots of x^+c=0 (mod
p).
V.
roots of
the reports,
of the papers by
p )
for
(mod
p)
congruent
to
of the roots is
The
fact
that
we
obtain
a
by
prime. Then
prime
(mod N)
powers of the
it is shown that the
sum, sum of products by
twos, threes,
or
even.
If
odd
x^=
1
(mod
5)
has
a
single
root.]
primitive
root
7-2,
so
that
x
continued
fraction.
In
and a
±r=x'/y', x /y ,.
continued
=
Vol.
1896;
modulus.
G.
F.
Meyer ^
gave
is
a
quadratic
non-residue
of
p
primitive
root
giving
to
±2
the
minimum
index
the
for which
for
p
[Chap, vii
in reverse
order those
etc.
in the
first line
are marked
=
the
/=0
(mod
p),
where
p
of
a^
p ),
where
p
is
a
prime
>2,
for
use
question of the number of satins. Given a^+l=0
(mod
p),
set
congruence.
B.
...+q^
S^,
B
x = 1
and
if
has the root
of roots is
d and
f
all integers n,
.o„_i]
p '/{2x).
is
X=x^'-'c(^^-'^^ '+i^/2 (mod p^).
G.
B.
Mathews'^^
(p.
53)
treated
p)
is
solvable
by
formulas.
Cf.
Legendre.-^^^
S.
Dickstein^^^
as
that,
if
p^),
 iCasopis,
Prag,
18,
1889,
97;
by
Lagrange
7+1
where the
a
quadratic
residue
of
p,
and
A:
2
—
it has an
index to the
p''),
for
the
primes
2,
3,
p
G.
Picou^*^
applied
to
=
[If 8a
P.
Bachmann^^
(pp.
344-351)
3-5-7)
by
modulo
7.
of
a
prime
p
divisor
p)
2 dividing
that k~
q.
Let
Un
By
p)
theory
prime
p,
where
r
t=ip^-2p^-'
^ML'iatermddiaire
dea
math.,
8,
1901,
162.
'^o*
Assoc,
Legendre's.
Call
M
the
to solve
directly x''=
£+3
£-1
(mod
p)
by
a
formula,
o
(mod
p).
We
can
express
S2m+\
in
the roots
p
and
a
y{nk-l)m/n
p *),
where
P,
Q,
respectively.]
L.
an
ilf
in
A.
Cunningham^^^
solved
x^=
—1
(mod
p),
where
p
certain
rational
be
2 Atti
*J.
Maximoff^^°
treated
—
by
assuming
that
does
not
c^?/^d,
in
wliich
each
first
(mod
p);
Q^
and
Q^'^
the
corresponding
Hence we
the numbers
Th.
Schonemann^^
powers
being
unity,
and
if
.,
=
the proposed
If
fx
are relatively
solutions
modulo
p
for modulus
of
solutions
following
paper
if
the
at
not ha\'ing
the root zero,
p
p
is that the
rank of the
solutions of the system
linear homogeneous
of
(2),
distinct
the sum of
order that
roots ?^0,
and found
of the rth
(2)
to
reduce
integer
x
if
the
former
are.
(mod
p'<)
p)
i=i
3=1
distinct
num-
bers.
Also,
^(n,
the number
\J/{n)
that
1,
2,
p)
does
right,
of
1,
the
congruences of degree n with no root
is
that
of
value of X the
\,
(2)
has
exactly
the
sign
of
differentiation
repre-
sents
conditions is
obtained. Con-
n
distinct
. .
.
ao'^o.
where
the
summation
which have no common
the
sum
of
=
i+j=0 (mod
by a
rational in
the
number
of
field
fix,
y)
x''-^+ . . .
(3),
with
a^
replaced
by
ap\
been made
firtesito
such
integer,
j^
is
a
p
con-
gruent
2n<p''
It follows
that there exist primitive roots a such that a^p^l if
e<p'' — L Any
p.
Every
are
All
congruences
of
the desired
Any congruence
of degree n has n real or imaginary roots. To find
them,
multiple root. The integral
x^'~^
. ..{x-^^' '-' '),
^0
(mod
p)
for
x
—
^*
m being the degree of F{x), and if the
function whose roots are
roots of F{x)
31, 1846,
a
divisor
of
p —
1,
and
if
^(a)
satisfies
modulo
p,
and
G(a)
function
of
Every irreducible
a
divisor
congruences
of
degree
n
modulis
p,
a,
the roots
(mod p-,
If is any integer and if F{x) has the
leading coefficient
by F(x)
modulo M.
factors modulo
f{x)
i^+1
(mod
p).
Then
n
powers
p,
p^,.
determines
/3
uniquely
with the
jS is
However,
be
given
a{i) with
that
Galois
had
given
no
imaginary roots of congruences
p,
M.
Let
p,
p,
which is
the generaliza-
tion of
Fermat's theorem.
Hence if A is prime to M, the above Unear con-
gruence
manner, and obtain
by P. In
where
the
summation
extends
over
Another
proof
is
based
modulo
p.
The
product
of
the
a,
6,
E.
Mathieu,^^
in
his
famous
. ..+(hzy '+hz+a},
over the roots of a^'^^a, while /i^ * =/i; and
(p.
302;
for
m
exposition, here avoided at
functions
of
degree
m
modulo
p.
A
function
a
proper
divisor
of
p
given degrees. If we
and
has
exception
the exponent I
and if we replace x by x^, where X is of the form indicated,
is
prime
2^~'^
g.
c.
a
product
of
2^ ~^
irreducible
functions.
by
p,
a:'' —
x
integers
modulo
p
we
so
in
if
we
irreducible
congruence
F(a:)
of func-
integers
t
is
a
root
congruences of
...+(
—
vi.
a
exponent n
and having
as coefficients rational functions of i, replace x by x^, where X
contains
only
the
number of
Frankfurt,
1867.
^«Trait^
des
substitutions,
1870,
14-18.
 Comptes
of degree
(^
—
by
X^.
Then
is
F„=nA
or
the
product
of
p
irreducible
functions
as
irreducible
factors
of
irreducible and
nor
p
Then
x^^
divisible
by
p
modulo
p
of
squares of
theorems
moduhs
p
(a
prime)
and
a
function
the product P
dividing
neither
p
/
modulo
p,
roots
of
roots
of
positive
integer
replaced
by
Let A
F{x). Then
formed of
generaliza-
tion
of
it is not
the form
a± hoi,
where co^
general
case
in
which
the
the
roots
of
all
of
whose
if B^O. In
z—^
is
irreducible in the GF[p ] if and only if n is not
divisible by
in
brief
no-
of
divisors
modulo
p
p.
Then
the
Let
of
[see
quadratic
residues].
L.
E.
GF[p''] of
L. Kronecker^°° treated congruences
defined
irreducible
cyclotomic
function
reducible
with
the Galois
(mod
p).
Since
Dickson^ ^''
purposes in
the
same
residue.
Thus
x^
systems
p,
in
regard
to
which
a
complete
the
residues
prime
to
p,
m, a
an/,
Jr{fn)
inverses
moduHs
m,
—1
when
m
is
a
power
Then
are
of
root of the GF[p^] if the
norm
of
of
p
process
of
prime
6n
a
quadratic
non-residue
of
fx,
as
binomial theorem satisfies
or does not
one
rational
of o-^
2q/
\ p
{2M
they are given
St.
Peters-
burg,
1894,
Ch.
I.
Cf.
Fortschritte
Math.,
25,
1893-4,
302-3.
or no
There
are
given
(p.
52)
representable
by
a
binary
congruences
by
in-
terpreting
(mod m) by use of Cardan's formula.
For
m
roots of
11^
defined
by
and the
cubic is seen to have real and two imaginary roots involving
i.
—
a
quadratic
residue,
is
one
roots of the equation
=
/),
(mod T),
viii
w=
I
7
I
if a/b is
(Serret's
Algebre,
ed.
5,
vol.
2,
466-8).
A
is
a
quadratic
—
other prime moduli
=0
last
mentioned
with
=
con-
gruence
has three real roots if one. To treat/(a)=a^+^a+Z,
add k/2 to
each root. For
the new roots,
relation.
the
corresponding
to
(2)
a
under
linear
diophantine
equations,
quadratic
is divisible
every
prime
same
roots,
but
equate
A[ni+
(mod
n,-n,).
Let
(ni.
ni,.
.nJ^AWJ-'
be
. .
.,
I'Argorithmie, Paris,
n's,
which
has
an
arbitrary
rational
value.
A.
Cauchy^
noted
that
if
Cauchy^^^
proved
that
a
congruence
/(x)
find
a
polynomial
</)(a;),
ducible
fraction
Genocchi proved that, if
the
p
i.
e.,
equation may
be interpreted
as the
and if
the resultant
one root in
can
s^, then
Giudice
case x'^-\-%Hx^-\-K=0
(mod /x),
are
a
/(a;)=0 (mod
the
remainders
on
Two
—
>««Bull. des
of odd
multiplicity, there
exist an
infinitude of
^''^Handbuch
modulo
m
taken
by
a
polynomial
in
and
with
Woodall
and
Creak,
a generalization.
.
^,=
(modp).
D.
power
p
of
the
prime
p
dividing
n
divisible
^
3, 1788,
p
of
solutions
m-zv'',
k
arbitrary.
Carmichael
power
of
the
prime
p
dividing
y,
and
evaluated
^SftSfc+^.
<r
by
p
all the
numbers
summed for
E. Catalan^^
86-7.
{a-\-^-\-pq-\-rs)\
D.
Andr^-^
lm+{n-l)k]k''-\
by
E.
number of powers of
Monthly,
progression [a part of
product
. ..(x+n-l)
is divisible
by n
and, if
x is
not divisible
power
of
p
dividing
(n
is
|
{(p-l)r+i)/p^-|[^]^+A:} (mod
of px=t
result.
«SitzunKsber.
Ak.
of
in
E.
elsewhere.
Integers.
of any r distinct
.
.
.,
n, r+1, the theorem shows that the product of any
n
prime
[a
power of A^. Hence, if N is prime to n ,
P is divisible
;'.
E, B.
Elliott^^ proved
of the
by
the
by the product
of
the differences of the squares of the first n odd numbers, multiplied
by
their
product.
a
prime,
(^n-/)
..=0
=
by
the
prime
p.
J.
Wolstenholme^^
proved
that
>
p
divides
.
.
the
prime
p
dividing
Proofs
by
Catalan,
ibid.,
1889,
19-22;
in
Mathews'
powers
p
and
2
of
and
2
which
=
case of
at a
all
positive
integers
r,
by
Lucas.
=
28,
1910,
26-28.
 Archiv
he
noted
that
(5)
Since
p
438;
ay^+hyz-\-cz^,
+
r.
Liou-
\'ille
(p.
267)
gave
p—
4^,
p
^x,
_p-3
(p-3)/4
S
symbol
and
Thus r^=
Thus, for
+
2<114,
(-l)'^2np2=22'».3.5.
p^).
He
x such that
will
be
designated
crk(n)
the
formula
(1)
fc=l
.,
.
.
.,
a(n—n)
p{x), but had no proof at
that
,„.
«
{o)
2j
expansion of
Vol. II.
of those divisors
.
.,
of m
where
C
isEuler's
a
product
of
a
primes dividing
(T(n)=nCn-C\ain-l)-C'2(T{n-2)-
(1),
. .
.,
A
complicated
.
.}
=
-
sum
of
^n/s.
Hence
''( )=iG]
This
sum
. . .
 Jour,
arguments
of
<r
are
2,
. . .
(Sc.
or
is
a
square;
a
square
. .
summed
for
m
Sd<T(d)=s(|y(r(d),
20M6m. Ac. Sc. St.
a
square;
=
=mS'-^.
The
number
ST(52'')(7(d)
which
implies
factor of the form 4)u+3 and if
we
2ciV,(5)
To
p
and proved
only
if
a
square
[cf.
condition
given
by
Bouniakowsky^°
N which
of exponents
.,
=
of the
This
recursion
formula
terms of
and
the
or even.
^riD^Mn/D^).
P'
the sum
of the
odd divisors.
s.i
pc
(-1)7
(t*«)'
n-
+^n-l
get a recursion
x/d is odd
<a/m of a.
. . .
=
where
of
partitions
he concluded
=
obtained dividing
by each
s„+(7(l)+(7(2)+...+(7(n)=n2.
and
.
.
..
solu-
tions
of
Ui.
he proved that
the number
divisors of any integer is log
2
; the
limit
for
^^Nouv.
Corresp.
Math.,
5,
1879,
296.
Diss.,
Berlin,
1881.
 Progr.,
Hamburg,
1881.
 Mathesis,
1,
1881,
99-102.
is
l/(p
we
numbers defined
by x,
form
(cf.
pp.
341-2
and,
for
a
are
multiples
of
If
(pp.
integers for
of
the
of the A;'s
erroneous.
For
instance,
proved
by
use
of
those divisors of
p.
220.
Next,
successive terms
. . .
a
. .
=
.
.
J.
Chr. Zeller^^ gave
J. W.
L. Glaisher®^
noted that,
we
suppress
the
undefined
term
(7(0),
+
of tables
=
which were
Let a,h, ...
prime
n,
A,(„)-.A,©-f.A,(|)
the latter
If
A'r(^)
is
the
sum
whose com-
sum
which
[whose
(Tr{i^),-
involving
as
coefficients
the
functions
f{d)
where d
ranges over the divisors of r, and 7n, n over all
pairs of integers
=
while
for
a
square;
2X(n)p,.
M
be
expressed
as
a
product
of
two
[(107),
(128)]
of
Liou\ille-^
and the
Xn\d,)
2m'(^)^(0
='r(r'),
+9A(4)A(n-4)+
where
b,?]=(7p(lK(2n-l)+(rp(3K(2n-3)
Glaisher'^^ proved
lower
[unstated]
and that
power other
of
the
remaining odd
divisors is
91,
II,
1885,
divisor
=1
(mod
=
which are not
di\'isors of x
the greatest square
the odd
di\'isors
. . .
=
the
second
equation
follows
from
. . .)=pSA^(n)x~,
for obtaining
(—n)th
1
—f(k)x^.
divisors,
if
is
even,
s„=
•
of
a
summation
formula,
number
L.
integers;
etc.
even,
and
powers all the
of the
 Messenp;er Math.,
sum
of x over the
than a
proof
8a:+5, 8a:+7)
$(l)+$(2)+
factors d,
=Si:{<j{i)<x{n-i)].
E.
inverse
function
is
x
3
^^
>6
of
<7
the
g.
(17)
S
1/^(772 —
random has
of
the
by
no
rih.
power
J.
is
not
a
.
=
. .
.
.
.
that
Next
he
proved
2d-\-7
=
1, 2,
next
>n;
for
A;
+(rm{n-4:)+(Tm{n-5)]-.
_{_
(^
_
2, 4,
E.
Lucas^^-
decompositions
of
1,2,.
in
not
expressible
sums of
for n odd, we
k-jr(T)—xi'm—p
are
Schroder
the
natural
are
x,2x,.
divisors of
1,
=r
which exceed
of n
6
is
a
the
of
(15).
M.
Lerch^^'
proved
that
2
remains
case
(17)
. .
where
v
where e
is of
n,
C
+6J,
where'
proved
(pp.
267-8)
(p.
345)
proved
(7)
employed
is
of
of
the
sum
of
function^'^
J{n)
2J{n)
inde-
pendently,
the
H{n)
odd
di\'isors, or more general
functions
(p.
I.
Giulini^^^
are
given
g{t)dt^
the last S
Tiixv/a),
where
divisors
of
the
for ^
n=0
n=0
Jacobi's
the coefficients
for
every
e
>0;
while
which
riri)
>2',
s
method used,
for the case in which n is an integral multiple of
a,
by
(11).
Proof
is
given
[n].
0.
Meissner^
squares divide
 »Arkiv
for
mat.,
cancellation
superior
limit
1
T(n)=S6„(fc),
k).
a;
of
where,
for
arbitrary algebraic
all or only
cited
here.
E.
proved
(p.
2223)
formulas.
S.
a
are
the
zeta
function
f
E.
Landau^^^
ix(d)
a2>ai>
v. If
xM=iMW.g)rg)=2.W.@x(0.
l ,n^j:r{n){x
—
is
J.
J.
exceeding
.,
fx^
or
ju^
fi^
and
inverses
the
quotient is odd, and the mean of the first last digit
of the quotient;
>
of
integers
of order
inferior to
contained
In
particular,
f{x)
divisible
by
the number
of repre-
first,
of
these
mul-
tiples,
intersection
J.
by
a
square
>1,
and
sion which are
L.
of
integers
1,
proved
that,
if
x
=
the
odd
integers
v'^2.
divisor
of b
.
.
.,
a series
composition of
not
divisible
formed of
r prime
V.A.Lebesgue^^notedthatthel.c.m.of a, .
rela-
such
J.
Neuberg^^
or
even.
T.
J.
operation on
the new
of
n
by
their
yl/{m)\p{n)
=<l>(mn).
of
m
or
the
number
a
polynomial
of
degree
ni
in
Ui,
712
in
^2,
Ci+TTix,
p
all
exponents
below
p
integers i, k
having t as
1897.
 BuU.
Ch.
III.
 Vorlesungen
mean
Q/iirH ^).
arbitrarily
chosen
integers
are
relatively
prime
is
=
g.
c.
the 1. c. m.
common
divisors
Di,
factors.
Let
1.
j,
prime to
is
residues
ka, kb is regarded
that X
and ax
just above
elsewhere.
On
the
g.
36,
1881,
is
stated
to
9.
ri,
r2,
more
than
two
digits,
follows
from
the
fact
that
lO-*
has
the
remainder
remainder on
by
p
Comm.,
5, 1767,
k
=
for the
divisibility
of
a
is added to the
is
listed
the decimal for
of
divisor
of
a
number
10n+
1
V.
of n
=*=
L.
L.
V.
di\isor
1, 3,
7 or
a
given
the
remainder
is.
p<100.
divisible
by
D,
where
divisible.
W.
Mantel
and
G.
A.
the di\isibility of
any base by a prime, the value of the coefficient
required
to
number
of
terms
of the period equals the length of the period of the periodic
fraction
t=^Au
Liljevalch^-''.]
factor
of
relatively prime, and
to
find
qi
so
that
qiX^^
1
(mod
p).
Let
a
the process, we get
multiple of
n. To
the
right,
decompose
N
into
. .
.
of N
A^ by
10
has
the
residue
q
same power
that in his
N.
F.
r digits of
only
if
N
divisors
of A'^ by
difference which is zero.
Quest. Educ. Times,
4,
1898-9,
241-3,
1
N.
Thus
of
sum
ao+
—
.
=
+
.
.
.,
A^
.
.
G.
Biase^^
7),
(mod
19).
above.
 Mathesis,
(3),
1,
1901,
197-8.
*^Rendiconti
x
to
the
baseB,
so
a
Given
.,
1889,
66,
107-10,
121-3.
de
Math.
(ed.,
Peano),
7,
1900-1,
42-52.
a .
1,
1911,
162-6.
*Paoletti,
by Barlow
divisibility
des
nombres,
Oxford,
1672,
a
numbers from 12 to
of 13th century) noted
numbers
all numbers up to
J. H. Rahn^ (Rhonius) gave
a
numbers, not
numbers, not
1,
EngHsh
table.
John
numbers
to
1000,
of all the
table was computed
of numbers to
10000
composite
3, 5,
of
numbers
10000.
Henri
Anjema-°
gave
of numbers to 10000.
Rallier des Ourmes-^ gave as if new the sieve of Eratosthenes, placing
3
called his attention to Brancker's^ table.
Lambert-^
gave
not
•Geometria Practica,
vol.
2,
1710
WaUis,
Opera,
2,
,511,
Leipzig,
1728.
 VoUst.
10
000,
Pisa,
1758.
depuis 1 jusqu'4
not divisible
Oberreit, von Stamford, Rosenthal,
constructing factor tables and
million, with plans for
Amstelodami,
1772.
26aNouv.
M6m.
sicher
zu
finden.
Nebst
Anwendung
der
Methode
auf
000,
copy
bekannten
Pellischen
2, 3,
the
856 000.
J. P.
^'Joh. Heinrich Lamberts deutscher
gelehrter Briefwechsel, herausgegeben von
der einfachen Factoren von
des nombres
and
1817
table is
the remainder
of hundreds
cell of
the nth
primes
<V,
factors
of
numbers
< 10000.
F.
J. HoiieP^
simples en
of
the
eight
volumes
of
2,
fourth,
fifth
and
sixth
milUons
presented
to
the
and hence
Rosenberg,
his
^^
H. Rosenberg),
phys.
et
nat.
100
000,
London,
1865.
••Jour.de
Math.,
(2),
11,
end of
of
factor
tables.
numbers.
Glaisher^^'
enumerated
=
0,
12,
24,
36, ...
60;
etc.
E.
Suchanek
continued
to
100
000
Simony's^^
primes
<N
by
Eratosthenes and to factoring.
base 900
and used
Sphinx-Oedipe,
5,
1910,
49-51.
^*La
the errata
7.
5
digits
nat., acad.
such
that
A^
table,
By a simple division
to
and the
The table thus
the
first
number
added,
90061,
the difference
n. They
ordinary
method
of
factoring
would
C.
F.
the form
squares,
let
g^
be
or
odd square just >r, according as r is even or odd, whence
c
h^-\-y
12, 1879, 715;
into
general
H. Hudson.^ Let
N be the
of
two
squares.
A.
described
the
method
of
are
small
.
,
(n^),
'Aux
C. Henry,
one
of
which
may
be
1.
the
—
1,
2,
example,
let
G
(a+kY-P,
where P
of P for
M.
between
10500
and
108000.
Kraitchik
h
number is composite
the
sum
of
the
sum
ad=f^bc,
a
sum
preceding
test.
=
1843,
313,
416-9.
 Novi
posthimious paper,
then
a;2
p.
Taking
p
serve
is a
prime
or
composite.
Th.
Similarly, if
in two
^y)
is
a
256.
3obTafebi
a=^x
five
a
a
prime
other
numbers
While
set D+n^^4Z) in
which
n
as
E.
Lucas^^
that
To factor
A^ seek
genus
has
only
means
of
deducing
complete
tables
from
the
given
in
factoring
a
E.
Lucas
gave
a
A, B are
subject.
T.
tested and
two factors of N,
{N
p.
used
48, 1889,
proved that
composite
unless
x^
Use made
On the line for
x when x makes
no
shaded
square.
Up
to
 Proc.
London
1, 2,
table
(pp.
291-2)
each
line
containing
four
given a
table of
6n=tl,
1,
in
or
9,
there
are
cases;
hence 20 cases in all. A. Niegemann^^ used the same method.
Anton
Niegemann^®
digits. Thus, if A76
formed
by
successive
additions.
G.
Speckmann^^
noted
is
=6
(mod
9).
made
an
extensive
a divisor lOw
be
a
and
120).
He
factored
and
Q
from
solution
the
form
1
fraction for
Uu^n+^-^X
N with respect to
and
proved
a ~H-D6^,
of x^ ^
modulo A'';
composite. But if it
Find
the
remainder
r
as before.
See
papers
14,
for a prime
186,
Leipzig,
1912.
6n'+l is
r odd, set
(Mem. de
and
double row for
=
9
if
Then
except
that,
C„
from
An
as
often
31,
19).
It
may
prove
iV=1829,
A^+1
forms
where
^j:
In
his
text,
consecutive integers,
and v^u'^S. Solving for
u, and
setting x
=
least
odd
factor
q
Jan.,
1912,
7-9,
A.
y
point
on
y
remarked
2 +^A:
odd
and stated that the
Sebastiano
641 of
proof of
^Oeuvres,
2,
25,
1640.
1777,
1779,
239.
is
a
prime.
An
anonymous
writer^^
stated
that
(1)
of Fermat's
by
means
of
the
series
term
divisible
by
is di\'isible
is
of F„
are of
it
the
we
^2 ^.
We
may
answer
the
of
the
modulo
F^
factor
K.
Multiply by i^
by
his^^
test,
before
D. Carmichael on
Phys.,
(2),
a
prime( ),
and
obtained
artificially
the
factor
(p.
167)
that
and F23
are com-
by
10^
—
2^^-37
2^^-397+1
3H
gave material
on possible
F^.
^^'^Math.
Quest.
a^=i=2ah+2b^.
a^
form (4n
q
chosen)
Euler^
treated
±
p^+g^,
and listed
all the
tabulated the
has a
the
form
2nx-{-l
or
divides
n
by
an
the linear
forms of
.
.
.,
forms
the
of
Jacobi's
=
and stated
divisible by n^,
that
2^ '''^
and
corrected
Plana.20
for various x's,
the
2 +l).
J. J.
other
than
has only
prime factors
powers
is
a
product
of
powers
or
2- ±
1
if
5
(1).
If
p
is
a
prime,
divisors
the
quotient
is
prime
to
m.
Sylvester^^
stated
.
.
R. W.
I
His formula
S'^^'+l,
relatively prime to
had
1 by use
p^3001,
A^.
90,
Vol. 28
F„
has
a
primitive
divisor
7^
1
except
forn
(mod
p^)
if
p
is
prime.
A.
>1,
F,(a)
has
a
prime
factor
Dickson®^
=
by
ex-
L. E. Dickson''^ factored w — 1 for various values of
n.
R.
D.
m, but not for 0<a;<m; in all other
cases
Q=
QXo-,
jS),
cases
(i)
c
a ^^=/3 /^
=
for
G.
Fonten^*°
stated
that,
if
p
is
a
is of the form kn+l, unless it is divisible
by
the
greatest
p
if
p
A.
Cunningham^
tabulated
a =tj8
prime
inves-
for
factors 2801
and gave an account
7^+2-7'+l, etc.
Factors^ *^ of
various trinomial
... in a
=
successive conver-
Ak
a
divisor
of
p+
1,
then
Be
is
divisible
by
p
for
e
not
a
if U, V are
that
is
a
quadratic
number of
division does not exceed five times the number of digits
in the
does not exceed
half
the
corresponding
Numbers,
p.
24,
Ex.
2.
the rank
of a
divisor
of
the
forms
By developing w„p and f„p in powers of
Un
and
of
sin
n
the
first
term
u^
p
law
are
is
stated
that
2^''
divisors.
A.
Genocchi^^
but
p
is
composite
if
+2
terms
is
by
p,
the
divisors
of
p
are
28Assoc.
frang.
avanc.
sc,
5,
1876,
61-67.
or
x^
by
p
is
5
2p
2±
Vs.
Then
if
p
of
3,
between sine and
In
sign
x^-\-Llf
of or
by
Lucas.'*
^Hhid.,
4,
1878,
1-8,
continuation
of
preceding.
^'lUd.,
pp.
33-40.
the cosine.
be
of
the
two
I
relatively
prime
integers
=
g
the
product
consecutive
terms
differs
terms differs from
pedal
triangle.
E.
series, then
—
.
. .
numbers which
has a
noted
that
Un
divides
of
units)
the
sum
recurrence with integral
generating equation
be roots
of unity.
math.,
5,
1898,
58.
series
defined
by
S.
series of
was
considered
by
E.
incomplete.
SeveraP^
discussed
the
 Zl/uoh+i
in relation
Wi
is a
odd.
between v^p
Pisano,
 Nature,
56,
1897,
10.
If 5x^±4
E.
Chicago,
1908,
109-113.
 L'interm6diaire
des
math.,
15,
1908,
248-9.
 OAmer.
Math.
Monthly,
15,
1908,
209.
soHandbuch.
particular, if
corrected form
Ui,
U2
Boutin^^
of Pisano's series.
E.
a
sum
of
a
depend on
. . .+iVT
the
general term of
a recurring series in the case of multiple roots by a more
direct process
Ill of this History.
that
C/„
Application is made to
terms
are
series.
' Analyse
•
+
(
in
series.
E.
Study
general term of
recurring series, exhibited
XVII
such
that,
d'Ocagne^^^
have
q
roots
in
J.
F
a
constant
of
d'Ocagne's^^^
and
papers
that of
and J. A.
Math.,
3,
1866,
finite when
gave the
pi,.
2 >iV-
L.
^m not
the primes
used
by
Kjonecker.^
Stormer-^^
gave
a
proof.
—
prime.
Legendre^^
noted
that
proof
that
mz-\-n
represents
infinitely
many
modulo
k.
Dirichlet^'*
extended
result
VJ?
of
terms,
 L'intermediaire
des
math.,
the following
the
case
m
a
prime):
If
the
theorem.
P.
the
primes
4>
procity theorem or
He supplemented the
•Improvements in
Legendre's
N
primes
gave
a
table
G.
tude
any given form
linear
same
number.
»8Sitzungsber.
Ak.
21,
1840,
98-100.
F.
Mertens^
gave
FOR Any Given
using the fact
F. Landry^^
A. Genocchi^^ proved
his
results.
L.
by
of
the
2»z-\-2''-^=i=l,
where
m
is
any
odd
square.
8n-l,
whereas
it
is
is
Euler's^°-
hence
the
primes
is suitable
x^+x+A
a
1843, 595,
x;
Uke-
is
a
prime
if
<p^,
that
of
<p(p-\-2),
and
no quotient
Two Primes.
be
even, n, n
is
a
Waring
by
O.
1891,
353.
a
sum
of
two
primes
in
—
of
3000 as
a sum
further
computations
enable
P(-l)=P(-3)=0.
G„ is
A. Cunningham^^^ verified
(2a)^ 2(2^=Fa),
17, p.
of carrying
log
p„
there
is
at
least
one
number
n.
It
the
n's for
p^
and
p/,+„
of
any of
the forms
verification
up
the
form
(5),
22,
II,
1913,
654-9.
not
11,
p
primes,
progression
unless
and
number
107+30/1 (n
=
=
furnishing
a
primality
test.
Tests
by
49;
forms
if
and
only
1,
3,
test has
similar
tests
for
integers
C-\-k^
is
not
+
the first s primes.
of
the
primes
2,
3,
composite
or
odd
composite
all zero,
then all the remaining coefficients, other than the first and last,
are
residues
?^0
of all odd
the product of all
Number of Primes
F.
Ro-
gel,^^°
J.
of
integers
of
the
first
6,
1872-3,
of
primes
<y/2N.
Also,
.p\
Lapj
LahpA
formula,
he
the
primes
Pi
p
and
q.
94,
II,
1886,
903-10.
which
are
of
one
in
which
x
is
formed
of
an
odd
number;
by a. He stated that
the
—
an
arbitrary-
6n+l
0(a:)=njl-^j,
is
^(1)+.
. .+^(m).
closed contour
enclosing the segment of the a;-axis from 1 to n and
narrow enough to
and
infinite
series,
a
and
/3.
composite or
prime. Hence
^x^ ~^^7(a: —
composite. Thus
^2a;( -i)y(a;'»-l)
and
of
Ch.
XL
G.
Andreoli^^
noted
^(x)
w is
taking
a:-axis in
natural
logarithms
—
at
least
one
to z.
between
two
consecutive
squares
<9-10^
x^l.
taken
Every prime
<rr
q^,
progression
Km-\-h.
  Comptes
Rendus
Paris,
116,
order,
prime >n.
A. A.
Markow^^*^ found
as
follows:
If
fx
is
the
greatest
5^
3,
and
i
between
p„
and
successive
integers
m,.
set
mi,.
r=oo
TO=oo
Several
asymptotically equal
posite numbers
of
deleted numbers
P„.
That
0(7r„).
stated
J.
x sufficiently
0-95Q
and
1-05Q.
of the
natural logarithms
8,
1849,
423-9.
Jour,
de
Math.,
19,
1854,
305-333.
»«Nouv.
Ann.
Math.,
10,
1851,
308-12.
P^tersbourg,
de
Bruxelles,
20,
II,
1896,
183-256.
which
3,
pp.
310-345.
m
of m
into set
S] all
T,
every
divisor
of
m,
imply,
respectively,
fim)
=2F(.,)
f{m)
=n/(p»), and/(p )
Cesaro^^ noted
if the sum
Xh{d)H{n/d),
cases. Then, if
2ju(d)
log
d=
one of
his appli-
of
consists
ai,.
13,
A. Berger^^''
^(1)
where
d8
a
determinant
F(l)
S
the unction defined
by the recursion
prime
—
N
N, e) constructed for
10000,
it
^(1)
to be
Fin,
rf)
a
b)
be
Let
4,
1902,
169-181.
values
of
fix)
n in [m],
numbers, to
Nazimov^ of
is
1
or
9,
1906,
408-14.
(Ch.
of /(n) if
by DF(n)
N. V.
we
have
for any
'^i
•
•
.,
on the right, d
^m
powers of the
on numerical integrals.
of n. He
of
fractional
[x]
can
number N
composed of
nated
while
the
sum
sum of
by
writing
its
digits
in
reverse
order
[Laisant*^].
P.
order. If
0.
A.
L.
Crelle^
quotient
be-
ing
equal
to
their
24^
or
42^.
The^^
three
digits
of
a
middle
digit,
the
^ffLadies'
Diary,
1811-12,
Quest.
12.31;
Leybourn,
Hence
base 10
its
digits
by
a,
and
number obtained
a
number
N
written
to
base
r,
then
P.
W.
G.
Cantor^^
employed
basal number k
e di Fis.
2, 2, 3,
same digits
as their
are the
 Nouv.
(2),
1,
1876,
403-11.
cubes.
C.
of the
by the nine
the
a
single
way
with ten zeros,
1896,
1897.
J.
order
and
nth
of
base
r;
A
of
m
the condition
remainder
on
division
by
3,
etc.
T.
P.
the
those
of
other
types.
(0,
1,
5
or
6)
(mod s'') for
where
N=
(/2 ~''
sets,
each
including
any
with
the
digits
of the
^th power of any one of these integers we can form
an
infinitude
where m is any
or
all
>0
nines,
digits, then
10''+1
NXH^BERS.
10
give
a
number
powers
for
/j,
digits.
E.
N.
Barisien^^
gave
and
digits as
of
the
10''(a-10' '+a-10 ^^'-^^+
(2),
21,
1912,
52-3.
the
digits.
Pairs'^
of
biquadrates,
the insertion
positive integers
so that
it is
of
two
and
Elem. Math.,
Those in
48
(28,
49)
Migne,
[4]
Montucla,
19
101)
Gegenbauer,
86,
93,
99
Genese,
78
Genty,
64
Gerhardt,
[59]
Glaisher,
99,
100
293, 295,