A BRIEF HISTORY OF THE INVESTIGATIONS ON MERSENNE NUMBERS AND
THE LATEST IMMENSE PRIMES
BY HORACE S. UHLER
THIS ARTICLE IS ADDRESSED TO THOSE VALUED READERS WHO PREFER
OR REQUIRE A SYNOPSIS OF FACTS BEFORE ALGEBRAIC PROOFS.
What are Mersenne numbers and why should any television fan abandon
this latest relaxation fad for neglecting the proper maintenance of
muscular tissue in order to consume any mental energy on something
as ancient, trite, and obvious as any kind of number? We shall
endeavor to give a succinct answer to the first part of this question
while abandoning the second part to the anthropologist, to the student
of the completely unreliable mammal often questionably referred to as
"homo sapiens."
It is convenient to call any number expressed by the form Mp = 2^p-1,
where p denotes a prime number, a Mersenne number regardless of the
fact that this philosopher extended his prediction only as far as
p = 257.
When[1] did Marin Mersenne live? He was born near Oizé (Sarthe) on
Sept. 8, 1588, and died in Paris on Sept. 1, 1648. Mersenne and
Descartes were fellow students at the Jesuit college of La Flèche.
In 1611 Mersenne joined the Minim Friars, and in 1620 he made
permanent residence in Paris at the convent of L'Annonciade. The
Minimi[2] (or Minims) were members of a Roman Catholic monastic order
founded in Italy in 1435 by St. Francis of Paula. This designation
was assumed as an expression of self-abasement. A portrait of Father
Mersenne is reproduced on page 72 of Oystein Ore's fascinating text
entitled Number Theory and Its History, (1948).
What interest attaches to integers of the form 2^p - 1, p prime?
Fundamentally because of their relation to the so-called perfect
numbers. A number is said to be perfect when it is equal to the sum
of all its integral divisors not including itself. For example 28 =
1 + 2 + 4 + 7 + 14 and is "perfect." Euclid (c 330 B.C.) proved that
a number satisfying the formulation (2^(p-1))(2^p - 1) is perfect
when the factor 2^p - 1 is a Mersenne prime. Obviously when the
number itself is included as its own divisor then the sum of the
divisors equals twice the perfect integer.
In this connection the following quotations are interesting and
informative. From pages 43 and 44 of Peter Barlow's book entitled
Theory of Numbers we quote "The difficulty, therefore, of finding
perfect numbers, arises from that of finding prime numbers, of the
form 2^n - 1, which is very laborious. Euler ascertained, that
2^31 - 1 = 2147483647 is a prime number; and this is the greatest at
present known to be such, and, consequently, the last of the above
perfect numbers, which depends upon this, is the greatest perfect
number known at present, and probably the greatest that ever will be
discovered; for, as they are merely curious without being useful, it
is not likely that any person will attempt to find one beyond it.
It may not be amiss to observe, that the reason for employing the
powers of 2 in this research, to the exclusion of all other numbers,
arises from this, that 2 is the only even prime; and, therefore, if
we made use of any other prime, as a, then (a^n - 1) would not be a
prime number; and, if we employed an even number, then the formula
above given would not truly express the sum of the divisors of N."
Barlow's rather derogatory opinions of the year 1811 are partially
shared by Edmund Landau.[3]
He says (in German): "This ancient idea
of the perfect number and its associated questions are not
especially important, I treat these matters only because by so doing
we shall immediately encounter two problems which are unsolved even
to-day: ... [1927]. Is the number of even perfect numbers infite? I
do not know. Is the number of odd perfect numbers infite? I don't
even know whether a single odd perfect number exists. Nevertheless
I beg the reader not to ponder long over these two questions; he or
she will learn about many more abstruse and promising problems during
the study of this work."
In marked contrast with the last opinion we read from the pen of
E. T. Bell:[4] "This raises the interesting and extremely difficult
question of finding those primes p which make 2^p - 1 prime. Any
help here would be appreciated by arithmeticians. For p = 2, 3, 5,
7, 13, 17, 19, 31, 61, 89, 107, 127, it is known that 2^p - 1 is
prime. Can anything general be stated? If so, whisper it to some
mathematician and you will probably be heard, from now till at least
the year 4000 A. D.)"
Greater human interest in Mersenne numbers developed from his
apparently off-hand prediction that the only values of p up to, and
inclusive of, 257 are 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.
Justifiable curiosity led to the investigation by many computers of
the prime or composite character of the numbers 2^p - 1 for all of
the 55 primes not exceeding p = 257. Comparison of the predicted
list with the correct set given in the preceding paragraph shows
that Mersenne made five mistakes.. He erroneously included p = 67,
257, and excluded p = 61, 89, 107. Since Mersenne made his
prediction in the year 1644 and the last one of the 55 candidates
2^p - 1 was finally examined in the year 1947[5] it is rather
astonishing that about 303 years were required to completely set
Mersenne right! (I wonder what his reactions were, if any.)
In critical vein attention may be called to a Special to the New
York Times, from Chicago, and published on Friday, March 27, 1936.
We quote: "A problem which has baffled mathematicians since the
time of Euclid, the finding of a perfect number with more than 19
digits, has been solved by Dr. Samuel I. Krieger." The 155-figure
value of 2^513 - 2^256 is then given correctly. But this equals
(2^256)(2^257 - 1) in which the M257 factor had been shown to be
composite by D. H. Lehmer and published in May 1931, nearly five
years earlier than Dr. Krieger's false claim to prominence. The
moral relative to the very best of newspapers should be obvious.
(By the way, the writer has sought in vain to find out the personal
history of D. Krieger. So haben der Krieger und die Zeitung gar
nichts "gekriegt.")
An extension of the playground belonging to the perfect numbers is
afforded by the multiply perfect numbers which are defined as
having the sum of their divisors equal to a multiple (2, 3, 4, . . . )
of the number itself. For example 672 = 2^5*3*7 is a number of this
kind since the sum of its 192 divisors equals 2*672. The multiple
may be called the class of the number. An enthusiastic amateur has
sent me (in 1949) one of his numbers of class 8, having 162 digits
and 13122 97817 86896 62976 divisors, inclusive of 1 and N.
The reader who desires full details concerning the names of the
investigators of the prime or composite character of Mersenne
numbers, of the factors when obtainable, of the dates and journals
of publication, etc., should refer to the characteristically
complete and reliable paper by R. C. Archibald in SCRIPTA
MATHEMATICA, v. 3, No. 2, p. 112-119, April 1935. Since Archibald's
paper practically exhausts the references up to the last date we
shall continue the present record after the year 1935.
At that time there were six values of p within the Mersenne range
which had not been investigated. They corresponded to p = 157, 167,
193, 199, 227, and 229. The work of E. Fauquembergue,[6] published
in 1914, established beyond question the primality of 2^127 - 1.
The interval of 21 years (1935-1914) seemed to give excellent
promise of a prime among the six virgin Mersenne numbers. These
considerations so overwhelmed the writer that he entered the arena
well armed and practiced but wholly ignorant of the strength of the
enemy. After tremendous labor caused by unreliable desk machines
and especially by an almost exaggerated estimate of the penalty of
an error of even one digit in any place, he eventually succeeded in
proving that all six of the Mersenne numbers are composite. It was
small consolation to realize that not a single prime occurred in,
the surprisingly long interval including the 24 prime values of p,
namely, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191,
193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, and 257. But
this is not all, the first sentence of a letter generously sent to
me by Lehmer and received on March 30, 1952, reads: "You will
doubtless be interested in the results on the SWAC on the night of
Jan. 30, namely that 2^521 - 1 and 2^607 - 1 are the next two
Mersenne primes. Since there are 42 prime p's between 257 and 521
the complete gap of composite numbers of the form 2^p - 1 (p-prime)
comprises 66 values. It was certainly unexpected that 12 prime
Mp's should occur in the first 31 values of p while not a single]
Mersenne prime or perfect number exists in the succeeding range of
66 prime p's. (Long live the "Queen and Servant of Science." E. T.
BELL.)
Why is it so difficult to test with desk machines a given number of
the form Mp? This is tantamount to asking precisely what the
procedure is. A brief outline of the high points of the method may
not be superfluous. The basic theorem, as perfected by D. H. Lehmer
from the pioneer results of Lucas, reads: The number Mp = 2^p - 1 is
a Prime if and only if it divides the (p - 1)-st term of the series
4, 14, 194, 37634, S[n],...where S[n] = S[n-1]^2 - 2. In non-
algobraic language we obtain the second element of this series by
squaring the first term and subtracting 2, i.e., 14 = 4^2 - 2. Next
we square the second term and subtract 2 to derive the third
element, i.e., 194 = 14^2 - 2, etc. Lehmer[7] says: "For numbers of
the form 2^(4p+-1)-1 Lucas advises the use of the series 3, 7, 47,
. . . This series does not give a necessary condition for primality,
while the series 4, 14, 194, . . . gives a necessary and sufficient
test for both 2^(4p+-1)-1." An equivalent corollary is this: The
number 2^p-1 is prime if and only if the (p - 2)-nd term of the
series 4, 14, 194, equals +-2^((p+1)/2) (modulo p). The point here
is that on repeated divisions by Mp the remainder for the (p - 1)-st
term must be zero and the remainder for the preceding division must
equal numerically 2 with the exponent (p + 1)/2 in order that
primality of 2^p - 1 shall exist. Obviously (p + 1)/2 will be an
integer because to be prime and greater than 2, p must be odd, p + 1
even, and hence (p + 1)/2 integral.
An extremely simple numerical illustration should suffice. Consider the
case where p = 7 and M[7] = 2^7 - 1 = 127. To have a non-zero quotient
we begin with the third term of the sequence 4, 14, 194, ... Divide 194
by 127 to obtain the remainder r3 = 67. (Don't bother about the
quotients.) Square 67 and subtract 2 to get 4487. Divide 4487 by 127
to produce the remainder r4 = 42. Square 42 and subtract 2 to find 1762.
Divide this by 127 using the quotient 14 (instead of 13) to produce the
smaller remainder r5 = -16. This satisfies the corollary -2^((p+1)/2) =
-2^((7+1)/2) = -2^4 = -16 hence 2^7 - 1 is prime. If we had gone one step
farther we should have squared 16 and subtracted 2 giving 254. Finally
division by 127 presents the remainder zero and thus satisifies the
necessary and sufficient requirement for primality. In this extremely
simple illustration we might have divided 127 into the fifth member of 4,
14, 194, ... that is, into 14163 17954 and obtained the remainder +111 or
111-127 = -16 at one stroke. Or again we may divide 127 into 2005 95654
68227 46114, which is the sixth term of the series (counting 4 as the first)
and find the remainder 0 at once. On the other hand this last procedure
is of no practical value because the higher terms of 4, 14, 194, ...
increase at a terrific rate[8]. For illustration, the tenth term contains
293 digits (roughly 3 X 10^2), the 226th term possesses very roughly
3 X 10^57 figures, and the 388th term boasts of about 1.8 X 10^116 digits.
Hence, in now obsolete practice one humbly commenced with the smallest term
of the series 4, 14, 194, which exceeded the divisor (modulus, modul) Mp.
Too much emphasis cannot be placed upon the fact that at long last the
courageous arithmetician has been emancipated forever from the slavery of
hand computation on Mersenne and analogous numbers by the appearance on
Earth of the magnificent digital calculating assemblages. And in point of
time these are still in their infancy. As stated in an earlier paragraph
the SWAC battery proved on the night of January 30, 1952, that 2^521 - 1
and 2^607 - 1 are Mersenne primes. Behold M521 = 68 64797 66013 06097
14981 90079 90813 93217 26943 53001 43305 40939 44634 59185 54318 33976
56052 12255 96406 61454 55497 72963 11391 48085 80371 21987 99971 66438
12574 02829 11150 57151 and M607 = 531 13799 28167 67098 68958 82065
52468 62732 95931 17727 03192 31994 44138 20040 35598 60852 24273 91625
02265 22928 56688 89329 48624 65010 15346 57933 76527 07239 40951 99787
66587 35194 38312 70835 39321 90317 28127, which PRIMES possess, respectively,
157 and 183 digits (when written on the base 10). The smallest term of the
series 4, 14, 194, ... which could be used as a dividend for the divisor just
written would be the 10th, which comprises 293 figures. Imagine dividing by
a number 183 digits long 596 or 597 times, squaring the remainders, etc.,
with a desk machine.
Finally and for completeness I squared 2^606 and multiplied by 2 to get
2^1213 with the object of computing the largest perfect number known to man
at this time. v14 = 2^(p-1)(2^p - 1) = (2^606) (2^607 - 1) = 2^1213 -
2^606 = 1 41053 78370 67120 69063 20795 80860 63189 88148 67435 14715 66783
88386 75999 95486 77426 52380 11410 41933 29037 69025 15619 50568 70982
93271 64087 72436 63700 87116 73126 81593 13652 48745 06524 39805 87729
62072 97446 72329 51666 58228 84692 68077 86652 87018 89208 67879 45147
83645 69313 92206 03706 95064 73607 35723 78695 17647 30552 66826 25328
48863 83715 07297 43244 63835 30005 31384 29460 29657 51433 68065 57075
95373 28128.
The SWAC digital computer uses the base 16 instead of the radix 10 as may
be inferred from the fact that the remainders kindly sent to the writer
by D. H. Lehmer were expressed in terms of 2^4. For example the SWAC gave
for the last remainder for M227 the value 3z 93458 0w7w0 45028 626z6 58002
91ux2 4xuw2 28vx7 zv7uy 36yv2
6y7wy where u == 10, v == 11, w == 12,
x == 13, y == 14, z == 15. This verifies exactly my (p - l)-th remainder
as first obtained on June 4, 1947, viz., 1071 24133 67508 18344 33653
76892 98433 16050 93930 31886 49512 37078 23311 35950. In other words the
same remainder reads (3)(16)^56 + (15)(16)^55 + (9)(16)^54 + ... + (7)(16)^2
+ (12)(16)^1 + (14)(16)^0, or, if the reader prefers, 3*2^224 + 15*2^220 +
9*2^216 + ... + 7*2^8 + 12*2^4 + 14*2^0, in the language of the SWAC and
1*10^69 + 0*10^68 + 7*10^67 + ... + 9*10^2 + 5*10^1 + 0*10^0 in the
language of the Collector of Internal Revenue.
The probability that a computer would obtain the (p - 1)-th remainder
from a series like 4, 14, 194, ... as zero when the Mersenne number was
actually composite is utterly negligible, although not impossible. On the
other hand, even with the utmost care and impeccable honesty it would be
possible ("human") to obtain an incorrect finite (p - 1)-th remainder with
the divisor Mp and to claim proof for the composite character of the
2^p - 1 number under investigation when the number is prime. A case in
point is that of Charles B. Barker who published a note indicating that he
had demonstrated the composite nature of M167[9]. The subsequent discovery
by D. H. Lehmer of the factor 2349023 for M167 established the facts that
this number is composite but that Barker's remainder from the Lucasian
sequence 3, 7, 47, ... was incorrect. In other words Barker's result was
illusory and he did not really prove anything about 2^167 - 1. Indeed the
same misfortune might have befallen the present writer for at least one of
the six then unexamined Mersenne numbers which he investigated and found
to have finite (p - 1)-th remainders. Hence it was a source of great
encouragement when the SWAC furnished the data (as of January 1952) which
completely verified all of his final test remainders. However the
discovery by Lehmer of the factors 2349023 and 1504073 for M167 and M229,
respectively, had previously substantiated two of my residues in the year
1946.
The only reason for relegating this paragraph to the end of the
historiette is that it refers chiefly to new prime numbers which are not
of the Mersenne type. From various sources,[10] including personal letters
from England, the following valuable information has been gleaned. A.
Ferrier proved that (2^148 + 1)/17 is prime. When written out on the base
ten we find the 44-digit number 2098 89366 57440 58648 61512 64256 61022
25938 63921. Using the EDSAC, Miller and Wheeler have demonstrated that
the expression k(2^127 - 1) + 1 generates a prime for each of the following
eleven values of k, namely, 114, 124, 388, 408, 498, 696, 738, 744, 780,
934, and 978. For visual reasons I have expanded the prime numbers
corresponding to the extreme values of k. For k = 114 the result has 41
digits and reads 1 93960 94914 49349 24174 12352 62361 07880 52879. For
978*M127 + 1 the decimalized representation 16 63980 77424 33890 86335
90183 03413 46554 01007 has 42 digits. Finally the M.-W. team can boast
of the 79-digit prime 180(2^127 - 1)^2 + 1 = 5210 64401 56792 28794 06069
43253 90955 85333 58984 83908 05645 83521 83851 01837 25557 35221.
Collecting, we record that of the 1951-52 record-breaking crop of 15
luscious prime numbers 5 have 41 digits each, 6 have 42 digits each, the
remaining 4 possess, respectively, 44, 79, 157, 183 digits.
ADDENDA
After the preceding manuscript had been received by the Editor-in-Chief
of SCRIPTA MATHEMATICA I became aware of the fact that valuable
information acquired subsequently from various sources concerning the work
with SWAC should be included in this paper provided that no unprofessional
duplication be thereby incurred. Hence I wrote to headquarters, that is
to my friend Professor D. H. Lehmer, Director of Research of the Institute
for Numerical Analysis, National Bureau of Standards, Los Angeles,
California. His answering letter was immediate and it contained without
reservation the following up-to-date minute details, as of June 4, 1952.
All Lucas tests that were performed earlier with hand machines have been
repeated on the SWAC. The machine definitively emphasized Barker's fiasco.
A few other disagreements will be investigated by Lehmer.
"The SWAC Mersenne Project consists in testing 2^p - 1 for p < 2309. Each
test is run twice with agreement, the second run being made at least a
week after the first. At present writing only two Mersenne primes have
been found, 2^521 - 1 and 2^607 - 1. The first run has reached p = 2011
and the second run p = 1307. The test for a given p takes approximately
(p/100)^3 seconds."
"The 'credit' for these results goes to the SWAC." Personal credit belongs
to Professor R. M. Robinson of the University of California, at Berkeley,
who designed this program.
As to prime factoring the SWAC has recently proved that "2^89 + 1 = 3 *
179 * 62020897 * 18584774046020617," a new valuable result.
In an earlier paper[11] we have recorded the numbers of digits in s12,
s13, and s14 to be, respectively, 1172, 2343, and 4686. 2^2309 and 2^2297
in order possess 696 and 692 digits. Since s11 has only 586 digits it is
clear that s12 would cover the present project for SWAC relative to
2^p - 1, p prime. Extensions of these data and their implications may be
derived from the following elaborations of the initial and final sequences
of figures in s12. The first 134 figures read 2231 39958 67897 90076
96037 96342 29578 85662 08710 40916 51298 31038 16096 83114 91946 22031
98934 07703 85772 14379 57260 31497 87541 60034 50340 10407 89215 and the
last 128 figures (ending at the unwritten decimal point) are 790 13713
78462 70954 69233 91035 93130 68881 74580 89003 06832 76492 57250 08680
49200 61619 79334 98686 55052 18272 48591 48886 69136 96655 34697 14434.
The joy (?) of computing the 910 intermediate figures may be left to some
electronic giant and its trainer.
My indebtedness to Professor Lehmer is appreciably enhanced and
gratefully acknowledged because of the following additional information
which he imparted to me in a letter dated June 16, 1952.
"The reason why p has to be less than 2309 has to do with the fact that
the SWAC has only 256 numbers of 36 binary digits each in its internal
memory. One half of this capacity is used for instructions. The other
half has to hold the 2p binary digits of U[n] and U[n+1] Hence 2p <=
128*36 so that p <= 2304. Of course you could not have known about these
details." Now the preceding paragraphs have come to a conclusion which
befits a brief account of the magnificent recent progress in the Mersenne
number field.
POST-ADDENDA
Paging Peter Barlow, Esq.! On August 21, 1952, the present author
received from Professor D. H. Lehmer a note containing the following very
exciting important facts: On June 25 the SWAC discovered that the
Mersenne-like number 2^1279 - 1 is prime. "We have tested this number
quite a number of times since. At high speed it takes less than 13 1/2
minutes. You may be interested in knowing that the 1277th term of the
sequence 4, 14, ... is congruent to -2^640 mod (2^1279 - 1)." The last
item may be looked upon as an additional check but the minus sign alone
is informative.
The writer immediately began to calculate on the base ten the exact
values of M1279 and the fifteenth perfect number v15. The value of 2^1279
was obtained by multiplying 2^1213 X 2^66 = 73786 97629 48382 06464. (vide
supra). The same 386-digit number had been worked out earlier by Dr. John
W. Wrench, Jr. Verbal comparison of the two independent results showed
them to be identical.
M1279 = 1 04079 32194 66439 90819 25240
32736 40855 38615 26224 72667 04805 31911 23504 03608 05967 33602 98012
23944 17323 24184 84242 16139 54281 00779 13835 66248 32346 49081 39906
60567 73207 62924 12950 93892 20345 77318 33496 61583 55047 29594 20547
68981 12116 93677 14754 84788 66962 50138 44382 60291 73234 88853 11160
82853 84165 85028 25560 46662 24831 89091 88018 47068 22220 31405 21026
69843 54887 32958 02887 80508 69736 18690 07147 20710 55570 31687 29087
The first calculation of v15 = (2^1278)(2^1279 - 1) = 2^2557 - 2^1278 --
the result of which is given below -- obviously necessitated the
evaluation of 2^2557. Since the latter datum, if published, was not
available to me, I formed the 770-digit product of 2^2222 X 2^335 because
these data were known to be absolutely correct. These powers of 2 possess
669 and 101 digits, respectively. The new result is
v15 = 54162 52628 43658 47412 65446 53743 91316 14085 64905 39031
69578 46039 20818 38720 69941 58534 85919 89999 21056 71992 19190 57390
08026 36461 59280 01382 76054 39746 26278 89030 57303 44550 58270 28395
13947 52077 69044 92443 14948 61729 43511 31262 80837 90493 04627 40681
71796 04658 67348 72099 25721 90569 46554 52996 29919 82343 10310 92624
24446 35477 89635 44148 13917 19816 44160 55867 88092 14788 66773 21398
75666 16247 14551 72696 43022 17554 28178 42548 17319 61195 16598 55553
57393 77559 23405 14622 23245 06715 97919 37573 72520 86087 82143 22052
22758 45375 52897 47625 61793 95176 62442 63144 80313 44693 50852 03657
58479 82475 36021 17288 04037 83048 60287 36212 59313 78999 49003 36673
94150 37472 24966 98402 82408 06042 10869 00776 70395 25923 18946 66273
61521 27756 03535 76470 79522 50173 85830 51710 28603 02123 48966 47851
36394 99289 04973 29214 51075 05979 91145 62215 19889 34576 49842 91328
The accuracy of this value of v15 deserves confidence because for every
one of the 86 sums-of-products-of-nonads involved in 2^335 X 2^2222 the
value was checked at once by an auxiliary modulus and later the definitive
product 2^2557 was shown to agree congruently with the product of the
separate factors on the basis of larger moduli. Incidentally the digits
0, 1, 2 ... 7, 8, 9 occur respectively in v15 67, 74, 84, 73, 84, 80, 76,
76, 69, 87 times.
YALE
UNIVERSITY
BIBLIOGRAPHY
1 Encyc. Brit., 14th ed., v. 15, p. 284.
2 Ibid., p. 541.
3 Edmund Landau, Elementare Zahlentheorie, Chelsea Publishing Co. (1946),
p. 18, 19.
4 E. T. Bell, Numerology (1933), p. 185, 186.
5 H. S. Uhler,
"On Mersenne's Number M227 and Cognate Data,
Bull. Am. Math. Soc., v. 54, no. 4 (April 1948) p. 378-380.
6 E. Fauquembergue, Sphinx OEdipe, v. 9, (June 1914) p. 85 103-105.
7 D. H. Lehmer, "An Extended Theory of Lucas' Functions," Thesis at Brown
University, (Dec. 26, 1929) p. 443-445.
8 H. S. Uhler, "The Magnitude of Higher Terms of the Lucasian Sequence 4,
14, 194, ...," Math. Tables and other Aids to Computation, v. 3, no. 22
(April 1948), p. 142, 143.
9 C. B. Barker,
"Proof That the Mersenne Number M167 Is Composite,"
Bull. Am. Math. Soc., v. 51 (1945) p. 389.
10 J. C. P. Miller and D. J. Wheeler,
"Large Prime Numbers,"
Nature, v. 168, (1951) p. 838.
J. C. P. Miller,
"Large Primes," Eureka,
no. 14 (1951) p. 10, 11.
D. H. Lehmer as Editor and Reviewer, M. T. A. C.,
v. 6, no. 37 (January 1952), p. 61.
11 H. S. Uhler,
"
Many-Figure Values of the Logarithms of the Year of Destiny and Other Constants,"
SCRIPTA MATHEMATICA, v. 17, nos. 3-4 (Sept.-Dec. 1951) p. 202-208.
Return to this
paper's entry
in Luke's Mersenne Library and Bibliography