1948-On_all_of_Mersenes_numbers_Particularly_M193-Uhler.pdf

(194 KB) Pobierz
102
MA
THEMA TICS: H. S. UHLER
PROC. N.
A. S.
ON
ALL
OF MERSENNE'S NUMBERS
PARTICULARLY
M193
By
H.
S.
UHLER
YALE
UNIVERSITY
Communicated
by
F. D.
Murnaghan,
January 15, 1948
On
November
27,
1947,
the author finished
investigating, by application
of the Lucasian
sequence
4, 14, 194,
37634, -..,
the
factorizability
of
Mersenne's
number
M193
=
2193
-
1
=
1255 42034 70773 36152 76715
78846
41533 28322
04710 88892
80690
25791.
The
192nd residue had the
value
542
45701
25193
90814 13211 43009
56802 04633
04970
79432 42801
51282
which, being
non-zero,
shows'
that
M193
is composite
in
character.
A
sufficiently
detailed
account
of the
procedure
followed
and
of
the
under-
lying
basic
theorem
may
be
found
in
the author's
first
paper2
on
Mersenne's
numbers.
Suffice
it
to
record
two
facts.
After the
date given above
the
data
of
the
entire
set
of
original work-strips
were
checked for the third
time with
an
auxiliary
modulus
in
conformity
with
the
condition
(Ml93qk
+
rk
+
2)
=
rk-12(mod
107
+
1),
where
q
and
r
denote
"quotient"
and
"remainder," respectively.
The
approximation
to
the
reciprocal of
M193
used in
the latest work
had the value
0.
(58 zeros)
79654
59555
66226
13851 44401 98883 85590
27955 52277
59630 93930
37006
37523
90143
41987 11093 10854
40700
94876
19334
41734
25085
25958
34273
96290
7.
*.
The check multiplication
gave
(M193)
(1/M193)a
=
1
+ 1.98 X
10-126.
With regard
to
the
55
numbers of the
form
2V
-
1
(where
p
is
a
prime
not
exceeding
257)
considered
by
Mersenne,
the
following
brief
report on
recent
progress
made
in
their
study
and
the
present
state
of
our
knowledge
may
be
timely because
of its definitive
character.
In
the
year
1935
there
remained3
six
Mersenne
numbers
which
had
not
been
investigated
with
respect
to
their
prime
or
composite
properties.
These
corresponded
to
p
=
157, 167,
193, 199,
227 and
229.
In
leisure
hours the
present
writer
began
the
investigation
of
M1i7
in
the
spring
of
the
year
1944 and
finished
the entire
set
with
M193
on
the
date
given
above.
These
Mp's
were
taken
up
in random order
in
the
vain
hope
of
discovering
a
prime
number
greater
than
2127
-
1.
More
specifically
the
final residues for
M157,
M16-, M229,
M199
and
M227
were
ob-
tained,
respectively,
bn
Aug.
11,
1944,
Dec.
2, 1944, Feb. 9, 1946,
July
27,
1946,
and
June
4,
1947.
(The
manuscript
on
M227
was
accepted
for
pub-
lication
in
the
Bulletin
of
the
Anerican
Mathematical
Society
on
July 17,
1947.)
Since
all
six
Mp's
were
found
to
be
composite
the
earlier
status
of
Mersenne's
remarks
has
not
been
changed
but
a
lacuna
in
our
knowledge
has
been
filled. The
next
extremely
difficult
step
will consist
in
the dis-
covery
of
all of
the
as
yet unknown
factors
of Mersenne's numbers.
Mersenne
said3
that the
only
values
of
p
not
greater
than
257
which
VOL.
34f
1948
V
,MA
THEMA
TICS: H.
S. VA
NDI
VER
103
make
Mp prime
are 2, 3, 5,
7,
13, 17, 19, 31, 67,
127
and
257. Comparison
of this
list
with
the
correct
data recorded
in
the top line
of
the
table
pre-
sented
below
shows that
Mersenne made five
mistakes. p
=
67
and
257
do not
yield prime
values
for Mp,
and p
=
61, 89
and
107
were not included
in his list of
special
primes.
With
reference
to
explicit factoring,
attention
should
be called
to
a
valuable paper4
by
Professor
D. H.
Lehmer
entitled "On
the
Factors of
1."
His
investigations
on
76 numbers unveiled eleven
factors
2n
which
fall
within
Mersenne's range.
Incidentally
two
of his
new
factors
confirmed
the present writer's
final
residues for
M167
and
M229.
p
CHARACTER
OF
M,
2,3,5,7,13,17,19,31,61,89,107,127
11,
23, 29, 37,
41,
43,
47,
53, 59, 67, 71, 73,
79,
113
151,
163,
173,
179, 181, 223, 233, 239, 251
83,
97,
131,
167,
191,
197,
211,
229
101, 103, 109,
137,
139,
149,
157, 193,
199,227,241,
257
'
Prime
Composite
and
fully
factored
Two or
more
prime
factors
found
Only
one
prime factor
known
Composite
but
no
factor
known
Lehmer,
D.
H.,
Jour.
London
Mlath.
Soc.,
9-10, 162-165
(1934-1935).
2
Uhler,
H.
S.,
Proc. Nat. A
cad.
Sci.,
30,
314-316
(1944).
8
Archibald,
R.
C., Scripta
Mlfathematica,
3, 112-119
(1935).
4
Lehmer,
D.
H.,
Bull.
Amer.
Mlfath.
Soc.,
53,
164-167
(1947).
NEW
TYPES
OF
CONGRUENCES
INVOLVING
BERNOULLI
NUMBERS
AND FERMAT'S
QUOTIENT
BY
H.
S.
VANDIVER
DEPARTMENT
OF
APPLIED MATHEMATICS,
UNIVERSITY
OF
TEXAS
Communicated
January 17, 1948
In
another
paper1
the
writer gave the
relation
r
(mb +
k)n
=
zQ()
(_1)ai
_____a
r
a-1
Sn(m,
k,
a);
r >
n;
(1)
where
we
define the
Bernoulli numbers bn by means of the recursion
formula
(b
+
1)"
=
bn;
n
>
1,
the left-hand member
being
expanded
in
full
and
b3
substituted
for
bt,
and
the
left-hand
member of
(1)
is
interpreted
in
the
same
way;
a-i
S,(m,
k,
a)
=
Z
s
=O
(im
+
k)";
00=
1,
m
and
k
are
any
integers
with
m
5
0.
In
the
present paper
we
shall
employ
(1)
as
well
as
other
known relations to
obtain
various congruences.
The
principal
results
proved seem to be
(5),
(6b),
(7)
and
(20).

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin