THE
AMERICAN
MATHEMATICAL MONTHLY
Entered at the Post-office at Springfield, Missouri, as second-class matter.
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VOL. XVIII. NOVEMBER, 1911 NO. 11.
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THE TENTH PERFECT NUMBER
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By R. E. POWERS, Denver Colorado
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A number which is equal to the sum of all its divisors is called a
"perfect number." Thus, the divisors of 6 are 1, 2, and 3, the sum of which
is equal to 6; the divisors of 28 are 1, 2, 4, 7, and 14, whose sum is 28.
Euclid (IX, 36) proved that if 2^p-1 is prime, then 2^(p-1)(2^p - 1) is a
perfect number, and no other perfect numbers are known. In 1644 Mersenne,
in the preface to his Cogitata Physico-Mathematica, stated, in effect, that
the only values of p not greater than 257 which make 2^p-1 prime are 2, 3,
5, 7, 13, 17, 19, 31, 67, 127, and 257. Regarding these "Mersenne's
Numbers" (2^p-1), W. W. Rouse Ball, in his Mathematical Recreations and
Essays (4th Edition, pages 262, 263, 269), says: "
"I assume that the number 67 is a misprint for 61. With this corre-
tion, we have no reason to doubt the truth of the statement, but it has not
been defnitely established. . . . Seelhoff showed that 2^p-1 is prime when
p=61. . . . and Cole gave the Factors when p=67. . . . One of the unsolved
riddles of higher arithmetic. . . . is the discovery of the method by which
Mersenne or his contemporaries determined values of p which make a number
of the form 2^p-1 a prime. . . . The riddle is still, after nearly 250 years
unsolved."
No exception to Mersenne's assertion (corrected by the substitution
of 61 for 67) is known at the present time. Below we show, however, that
2^89-1 is a prime number, contrary to his statement.
Following is a list of Mersenne's Numbers thus far proved to be prime,
with the corresponding perfect numbers:
p 2^p-1 Perfect Numbers
2 3 6
3 7 28
5 31 496
7 127 8,128
13 8,191 33,550,336
17 131,071 8,589,869,056
19 524,287 137,438,691,328
31 2,147,483,647 (19 digits)
61 2,305,843,009,213,693,951 (37 digits)
To these must now be added the prime number 2^89-1, so that the
tenth perfect number is 2^88(2^89-1), or
191561942608236107294793378084303638130997321548169216
(it is known that 2^p-1 is composite for all other values of p not greater
than 100).
In his Théorie des Nombres, page 376, Lucas says: "Nous pensons
avoir démontré par de très longs calculs qu'il n'existe pas de nombres par-
fait pour p=67 et p=89." While this result has since been verified for
p=67, the opinion has been expressed that also the case p=89 needed an in-
dependent examination. The result here shown that 2^89-1 is a prime
is therefore in conflict with Lucas'computation. The same writer, in an
article entitled "Théorie des Fonctions Numériques Simplement Périodques,"
Section XXIX, in the American Journal of Mathematics, Volume 1 (1878),
proved the following remarkable theorem (thw theorem appears on page 316
of the volume):
"If P=2^(4q+1)-1, and we form the series of residues (modulo P)
4, 14, 194, 37634, ...,
each of which is equal to the square of the preceding, diminished by two
units: the number P is composite if none of the 4q+1 first residues is equal
to 0; P is prime if the first residue 0 lies between the 2qth and the
(4q+1)th term."
Applying the above theorem to the number 2^89-1, and denoting the
terms of the series by L(1), L(2), L(3), ..., we found the following
residues (modulo 2^89-1):
m L(m)
1 4
2 14
3 194
. . . . . . . .
10 -115,113,975,804,653,882,052,836,464
20 36,000,517,785,442,762,303,479,300
30 -204,144,540,641,167,292,618,604,303
40 -126,791,709,316,676,382,795,042,761
50 -90,990,560,635,837,660,454,542,648
60 -206,308,592,424,355,282,693,419,690
70 99,498,791,857,820,493,810,407,653
80 269,783,273,665,984,523,074,966,550
. . . . . . . .
86 -309,403,333,482,440,150,628,882,422
87 -35,184,372,088,832
88 0
Since the first (and only) residue 0 occurs at the 88th term of the
above the series, it follows, from the foregoing theorem, that 2^89-1, or
618,970,019,642,690,137,449,562,111
is a PRIME NUMBER.
As M. Lucas points out, his method used above is free from any un-
certainty as to the accuracy of the conclusion that the number under consid-
eration is prime, in case our attempt to arrive at the residue 0 meets with
success, since an error in calculating any term of the series would have the
effect of preventing the appearance of the residue 0. We would add that,
denoting the number 2^89-1 by N, we have verified that
3^(N-1)-1 is divisible by N,
which is in accordance with Fermat's well-known theorem.
Denver, Colorado, June, 1911.