Download 207 following values of p

207
TECHNICAL NOTES AND SHORT PAPERS
1. H. Lamb, "On the propagation
of tremors
over the surface of an elastic solid," Roy. Soc.
London, Phil. Trans., ser. A, v. 203, 1904.
2. C. L. Pekeris, "The seismic surface pulse," Nat. Acad. Sei., Proc, v. 41, 1955, p. 469-480.
3. NBS Applied
Mathematics
Series, 37, Tables of Functions
and of Zeros of Functions,
U. S.
Gov. Printing Office, Washington, D. C, 1954. {MTAC, v. 10, 1956, Rev. 104, p. 249-250.]
4. P. Davis & P. Rabinowitz,
"Abscissas and weights for Gaussian quadratures
of high order,"
National Bureau of Standards, Jn. of Research,v. 56, 1956,p. 35-37. {MTAC, v. 11,1957, Rev. 84.]
Mersenne
Numbers
By Hans Riesel
During 1957 the author of this note had the opportunity
of running the
Swedish electronic digital computer BESK in order to examine Mersenne numbers. The intention of the author's investigation on the BESK was to check some
known results, and to examine some Mersenne numbers not previously examined.
Mersenne numbers are numbers Mp = 2P — 1, where p is a prime. See QlJ,
which contains a more complete list of references. The Mersenne numbers have
attained interest in connection with digital computers because there is a simple
test to decide whether they are prime or composite. This is Lucas's test [TJ.
Furthermore
the number 2P~1MP is a perfect number, if Mp is a prime, and all
known perfect numbers are of this form.
In the beginning of 1957 a program for testing the primeness of the Mersenne
numbers on the BESK was worked out by the author. This program, using
Lucas's test, works for all p < 10000. As a test, this program was run for the
following values of p: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279,
2203, and 2281. The result was that the numbers Mp are prime for these values
of p, thus confirming known results.
After these tests, values of p > 2300 were to be tested. These values had not
been tested before. But since the testing of Mp for one value of p of this order
takes several hours on the BESK, a special program for calculating the smallest
factor of Mp, if this factor is <10 -220 = 104 85760, was worked out. This special
program is based on the following well-known theorem :
All prime factors q of MP (p > 2) are of the form q = 2kp + 1, and of one
of the two forms q = SI ± 1.
The proof of the theorem is quite simple. If g is a factor of Mp, 2P = 1 (mod g).
Since p is a prime, since all numbers n for which 2n = 1 (mod g) form a module,
and since 2P = 1 (modg), this module consists of all integral multiples of the
prime p. Now 2«-1 = 1 (mod q) if g is a prime, and hence g — 1 is a multiple of p,
and in fact an even multiple (since g must be an odd number). This is the first
part of the theorem:
immediately
the solution
g — 1 = 2kp (k = 1,2,3,
•••)• The second part follows
from the theory of quadratic
residues. Since x2 = 2 (mod g) has
x = 2(p+1)/2(modg), we see that 2 is a quadratic residue modg,
hence g = ± 1 (mod 8).
By
of Mp
residue
found,
the above mentioned special program for small factors of Mp
(mod g) for all primes g of the theorem were now calculated.
was =0, the factor g was printed out. When no factor g <
the BESK turned to the next value of p. This program was
Received 8 November 1957.
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the values
When this
10 -220 was
run for all
208
TECHNICAL NOTES AND SHORT PAPERS
values of p < 10000. The running time on the BESK for one value of p lay
between 0 and 2 minutes, depending on the size of the smallest factor. These
smallest factors are given below in a table. The known Mersenne primes are also
included in the table. Finally, Cole's factor of AÍ67, and Robinson's
factors of
Afio» and Af167,though greater than 10-220, have been printed in the table. On
checking the values against other sources, a misprint in Archibald's
note £2]
was detected. This misprint, which concerns the value of the smallest factor of
Afi63, seems to have come from Kraitchik
[3], p. 24 and p. 92. As a further
check, all factors in our table have been looked up in Lehmer's Prime Tables,
except a few, which are too large, and a separate calculation was made, to check
that they are of the form 2kp + 1. Since, however, there are disturbances
in
digital computers,
it is not absolutely sure that all these numbers really are
factors of the corresponding
Mersenne numbers Afp. Those primes p, for which
no factor < 10 -220 of Mp was found, are, except the known Mersenne primes,
omitted in the table.
When this table had been calculated, the BESK examined the omitted values
of p with Lucas's test. This was done for all values of p between 2300 and 3300.
Since the available running time for testing Mersenne numbers on the BESK
was limited, every value of p was tested only once. The final remainder, see rjlj,
was printed out in hexadecimal form. On September 8th, 1957, a run indicated
that the number Af32i7 is prime. This result was repeated on September 12th.
All other numbers tested turned out to be composite ; however, this result could
be false, since the running time is very long. The testing of Af32i7 took about
5h 30m on the BESK, for one run.
A table of the smallest
factor of 2" — 1, p prime follows. Primes
2P — 1 are
indicated by a dash.
Ormangsgatan 67C
Stockholm-Vallingby, Sweden
1. R. M. Robinson,
842-846.
Mersenne
and Fermât
numbers,
Amer. Math.
Soc, Proc,
v. 5, p.
2. R. C. Archibald,
"Mersenne's numbers," Scripta Mathematica, v. 3, 1935, p. 112-119.
3. M. Kraitchik,
Recherches sur la Theorie des Nombres, Gauthier-Villars,
Paris, 1952.
Table of Factors of 2" - 1 : f £
2
—
3
—
5
—
7
—
11
23
13
—
17
19
37
223
43
431
71
2 28479
107
—
157
8521 33201
29
233
53
6361
31
—
41
13367
67
1937 07721
97
11447
151
18121
23
47
47
2351
61
73
439
109
7459 88807
79
2687
113
3391
59
1 79951
83
167
127
163
1 50287
167
23 49023
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
173
7 30753
89
—
131
263
179
359
TECHNICAL
181
43441
233
1399
283
9623
367
12479
449
12 56303
521
593
1 04369
643
31 89281
757
98 15263
839
26849
883
8831
953
3 43081
1021
40841
1091
87281
1153
2 67497
1223
2447
1321
7927
1433
20063
1481
71089
1543
1 01839
1667
13337
NOTES AND SHORT PAPERS
Table of
of 2" -
191
383
239
479
311
53 44847
383
14 40847
461
2767
547
5471
601
3607
659
1319
761
4567
857
6857
907
11 70031
967
23209
1031
2063
1093
43721
1181
47 42897
1229
36871
1327
27 30967
1439
2879
1489
71473
1559
3119
197
7487
251
503
317
9511
397
1693
10159
2383
463
11113
557
3343
607
683
1367
773
68 64241
859
72 15601
911
1823
977
8 67577
1033
1 96271
1097
9 80719
1187
2 56393
1231
5 31793
1361
8167
1447
57881
1499
2999
1583
3167
1697
12 35417
r
_
'factor
211
15193
263
23671
337
18199
419
839
487
4871
571
5711
617
59233
701
223
18287
277
11 21297
353
9 31921
431
863
491
983
577
3463
619
1 10183
719
1439
827
66161
877
35081
937
28111
1013
6079
1049
33569
1117
53617
1201
57649
1279
7 96337
811
3 26023
863
82 58911
929
13007
1009
34 54817
1039
50 80711
1103
2207
1193
1 21687
1249
97423
1367
1381
10937
8287
1451
1453
2903
8719
1511
1523
3023 25 22089
1607
1609
28927 23 94193
1709
1721
1 75543
3 79399
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
229
15 04073
281
80929
359
719
443
887
499
20959
587
5 54129
641
35897
743
1487
829
72953
881
26431
35
3
104
16
941
7529
1019
2039
1051
75503
1129
33871
1213
27511
1303
44849
1423
99063
1459
14591
1531
88799
1663
16631
1723
17231
TECHNICAL
NICAL NOTES ANE
AND SHORT PAPERS
Table of Factors
Factors of 2P -— 1 : ,
1741
10 02817
1847
33247
1931
3863
2003
4007
2081
2 66369
2179
2 48407
2297
57 65471
2399
4799
2539
25391
2617
78511
2689
71 58199
2767
6 25343
2903
5807
3001
32 17073
3121
11 23561
3257
97711
3359
6719
3499
35 82977
3539
7079
3623
7247
3779
7559
10
1
21
72
1777
10663
1861
23551
1973
22327
2011
71881
2083
69671
2203
2333
3 12623
2411
19289
2543
5087
2621
15727
2699
5399
2819
5639
2909
1 10543
3011
20 23393
3137
20 01407
3299
6599
3361
5 57927
3511
35111
3541
7 64857
3659
41 05399
3793
60689
t —Continued
1789
1801
1811
39359
28817
3623
1871
1877
1879
14969
15017
6 05039
1987
1993
1997
67559
11959
3 95407
2039
2063
2017
4079
4127
93 38711
2111
2113
2131
3 41983 22 31329 32 73217
2213
2251
2207
4 00679
1 23593
53113
2339
2351
2389
4679
4703
71671
2417
2447
2459
4919
14503 81 63193
2591
2593
2549
15559
43 63889
7 66937
2677
2657
2663
63913
3 64073
37199
2741
2707
2711
82231
1 73249
5 85577
2857
2837
2843
1 42151 36 79817
22697
2953
2963
2939
88591
5927
5879
3041
3023
3037
24329
18223
6047
3217
3181
3163
19087
4 55473
3329
3323
3319
26633
33191 23 12809
3457
3433
3391
20743
2 98409 44 42303
3527
3529
3517
1 05871
63487
5 62721
3571
3581
3557
64279
21487
4 26841
3733
3719
3701
1 11031 43 51231 13 14017
3823
3803
3797
7607 89 68759
91129
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1823
1 20319
1913
63 01423
1999
18 07097
2069
3 26903
2141
3 89663
2281
2393
33503
2531
7 89673
2609
36527
2687
1 98839
2753
47 95727
2897
17383
2999
6 71777
3119
24953
3221
6 44201
3347
26777
3491
6983
3533
1 90783
3593
21559
3767
30137
3851
7703
TECHNICAL NOTES AND SHORT PAPERS
Table of
3853
33 90641
3931
26 18047
4057
97369
4211
8423
4289
34313
4457
16 31263
4597
27583
4723
29 09369
4861
29167
4957
1 48711
5021
40169
5167
65 31089
5231
10463
5303
10607
5431
54311
5519
6 62281
5639
11279
5791
9 26561
5903
11807
6113
25 30783
6199
61991
3863
7727
3967
56
21
8
58
6
1
3
9
3
20
8
6
1
63473
4073
69617
4217
84407
4297
93777
4483
54799
4603
26009
4759
14217
4871
9743
4967
27823
5039
10079
5171
10343
5233
94271
5347
10127
5437
87809
5521
88337
5683
75183
5801
34807
5939
17657
6131
12263
6203
48873
r
211
_
>rsof 2" -
' factor
3907
5 31353
3989
1 91473
4099
73783
4229
3 29863
4373
61223
4513
1 35391
4639
1 94839
4787
1 14889
4903
49031
4993
79889
5051
10103
5179
2 48593
5237
40 22017
5393
32359
5441
36 56353
5531
9 40271
5711
11423
5807
1 39369
6011
2 88529
6143
8 84593
6263
12527
3917
4 07369
4013
1 20391
4129
15 85537
4271
1
8543
4441
26647
4549
1 36471
4657
5
11 45623
4801
28807
4933
29599
5003
2
10007
5101
81 10591
5197
1
31183
5279
10559
5413
53 26393
5483
25
8 88247
5557
5563
33343
5 34049
5717
5743
34303
5 43217
5843
5849
7 12847 41 05999
6029
6067
84407
2 06279
6173
6163
5 91649
37039
6277
6287
37663 50 29601
3911
7823
4001
24007
4127
74287
4243
01833
4391
8783
4517
27103
4649
95073
4793
67103
4919
9839
4999
09959
5087
40697
5189
55671
5261
42089
5399
10799
5449
17439
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
3923
2 19689
4019
8039
4153
91367
4273
25639
4447
71153
4561
72977
4691
5 72303
4813
28879
4943
9887
5011
80177
5107
51071
5227
11 29033
5297
44 49481
5417
4 33361
5501
6 93127
5623
11 35847
5779
9 24641
5861
46889
6101
79 06897
6197
2 97457
6311
36 47759
212
TECHNICAL
NOTES AND SHORT PAPERS
Table of Factors of 2P — 1 : factor
6317
6 94871
6389
1 91671
6491
12983
6569
66 21553
6719
12 90049
6883
18 85943
6967
10 72919
7057
6 21017
7207
1 72969
7487
1 79689
7591
18 97751
7691
15383
7853
47119
8017
1 28273
8171
8 00759
8243
16487
8377
50263
8461
14 04527
8629
2 58871
8719
28 77271
8803
28 34567
8951
17903
6323
12647
6397
19 44689
6521
48 90751
6607
80 34113
6737
7 41071
6899
13799
6977
41863
7079
14159
7211
14423
7529
1 05407
7639
5 50009
7741
89 64079
7873
11 80951
8059
70 43567
8179
43 18513
8269
7 27673
8387
92 76023
8467
2 03209
8663
17327
8731
25 14529
8807
2 28983
8963
6 63263
-Continued
6343
6353
90 45119
38119
6421
6449
56 89007
51593
6529
6551
7 31249
13103
6637
6673
52 03409 67 66423
6781
6803
1 08497
2 17697
6917
6947
1 80623
55337
6983
6997
13967 21 69071
7103
7129
14207 18 67799
7297
7411
18 68033 10 22719
7537
7561
7 23553
5 89759
7643
7673
15287
1 84153
7759
7789
1 39663
8 72369
7883
7901
15767
47407
8101
8111
7 12889
16223
8219
8221
19 88999
9 20753
8293
8273
1 99033
10 42399
8423
8419
4 88303
3 53767
8501
8513
2 72417
6 80081
8669
8693
2 60791
26 18039
8741
8737
69929
6 81487
8839
8821
4 05767 65 23183
9007
9001
90071
15 12169
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
6367
63671
6473
27 96337
6553
14 54767
6691
3 21169
6841
41047
6949
68 79511
7039
12 52943
7151
14303
7417
1 18673
7573
45439
7681
39 94121
7823
15647
7919
63353
8161
67 08343
8231
2 14007
8317
3 82583
8429
67433
8597
4 81433
8699
4 17553
8761
1 92743
8867
70937
9029
7 22321
6373
38239
6481
6 22177
6563
13127
6703
4 42399
6871
2 88583
6959
55673
7043
14087
7193
54 09137
7459
1 34263
7589
2 88383
7687
4 45847
7841
23 67983
8011
80111
8167
31 36129
8237
10 54337
8353
50119
8431
8 09377
8599
85991
8713
16 03193
8783
2 28359
8929
1 96439
9059
18119
TECHNICAL
9109
49 55297
9221
7 19239
9343
1 49489
9421
85 73111
9511
95111
9791
19583
9883
1 58129
NOTES AND SHORT PAPERS
p - -Continued
Table of Factors of 2" — 1-' factor
9127
9137
9157
9173
1 46033
10 05071 30 76753
4 95343
9311
9323
9337
9283
59 41121
14 15273
8 95009 26 14361
9371
9391
9397
9403
18743
93911
2 25529 25 76423
9491
9431
9461
9479
18959
6 79033
75689
5 31497
9521
9539
9601
9613
19079 22 85039
3 61799
57679
9811
9829
9833
9851
78809
77 70313
07689 45 62513
9949
9973
80 98487
99191
A Computation
of Some Bi-Quadratic
213
9181
13 77151
9341
74729
9419
18839
9497
5 31833
9619
6 15617
9859
11 04209
Class Numbers
By Harvey Conn
A fascinating chapter in computational
number theory began when Lagrange
showed that every positive integer is representable
as the sum of at most
four perfect squares [lj. Clearly three would not suffice in every case, as
7 = 22 + l2 + l2 + l2 would be an exception ; nevertheless, the problem of expressing some positive integer n as the sum of at most three squares soon achieved
a very special role. For, Gauss showed that r(w), the number of such representations, is connected in a very simple way with the much studied (intrinsically
positive) class number, h, of the field generated
by -\¡—n. Specifically for n
square-free (and n ^ 1,3 where h = 1),
(D
rin)
=
where g = 12 for n = 1, 2, 5, 6 and g = 24 for n = 3 (mod 8). Thus we could
conclude the existence of at least one such representation
for the indicated n.
Gauss and later, Kronecker, reversed the direction of these equations by making
large scale tabulations
of h from rin), although, unfortunately,
no location for
Kronecker's
alleged tabulation
(for odd n up to 10,000) seems to exist in the
literature.
In tallying the representation
n = Xi2 + x22 + x32 it might be noted
that one must count each ordered triple (xi, x2, x3) of positive, negative, or zero
integers as a separate unit, so that as much as 23-3! = 48 could be contributed
to rin) when such a decomposition
into squares is expressed as triples.
In more recent times, the representation
theory was extended to integers in
the field k generated by V5, i.e., to the quantities ß = (a + ¿>V5)/2 where a and b
are of the same parity. Here we seek to represent, necessarily, only those integers
fi which are positive together with their conjugate (i.e., totally positive). Thus,
e.g., a > |ôa/5| > 0. The special surd V5 must be used because then, as Götzky
Received 8 January 1958.
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