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[PDF page 245] CH. IX] MERSENNE’S NUMBERS. 229 That N is prime whenp = 61 had been conjectured by Landry and in 1886 a demonstration was offered by Seelhoff*. His demonstration is open to criticism, but the fact has been verified by others†, and may be accepted as proved. That N is composite when p = 89 seems to have been verified by Lucas‡in 1891, but the demonstration has not been published, nor have the actual factors been discovered. That 7487 is a factor ofN when p = 197 was shown by A.J.C. Cun- ningham in 1895§. That N is not prime whenp = 67 seems to have been verified by Lucas∥ in 1876 and 1877. The composite nature of1 N = 2p − 1 when p = 67 was confirmed by E. Fauquembergue¶, and was also implied by Lucas** in 1891. The factors were given by F.N. Cole††in 1903. Bickmore in the memoir‡‡cited below showed that1433 is another factor of N if p = 179; and that 1913 and 5737 are other factors of N if p = 239 . I turn next to consider the methods by which these results can be obtained. It is impossible to believe that the statement made by Mersenne rested on an empirical conjecture, but the riddle as to how it was discovered is still, after nearly 250 years, unsolved. I cannot offer any solution of the riddle. But it may be interesting to indicate some ways which have been used in attacking the problem. * P.H.H. Seelhoff, Zeitschrift für Mathematik und Physik, 1886, vol.xxxi, p. 178. † See Weber-Wellstein, Encyclopaedie der Elementar-Mathematik, p. 48; and F.N. Cole, Bulletin of the American Mathematical Society, December, 1903, p. 136. ‡ Théorie des Nombres, Paris, 1891, p. 376. § Proceedings of the London Mathematical Society, March 14, 1895, vol. xxvi, p. 261. ∥ Sur la Théorie des Nombres Premiers, Turin, 1876, p. 11, quoted by Lieut- Colonel A.J.C. Cunningham,Proceedings of the London Mathematical Society, Nov. 14, 1895, vol.xxvii, p. 54, andRecherches sur les Ouvrages de Léonard de Pise, Rome, 1877, p. 26. ¶ L’Intermédiaire des mathématiciens, Paris, Sept. 1894, vol.i, p. 148. ** Théorie des Nombres, Paris, 1891, p. 376. ††On the Factoring of Large Numbers, Bulletin of the American Mathematical Society, December, 1903, pp. 134–137. ‡‡C.E. Bickmore,Messenger of Mathematics, Cambridge, 1895, vol.xxv, p. 19. 1. Corrected: originally N 2p = 1

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[PDF page 246] 230 MERSENNE’S NUMBERS. [CH. IX The object is to find a prime divisorq (other thanN and 1) of a number N when N is of the form2p −1 and p is a prime. It can be easily shown that q must be of the form2pt + 1. Also q must be of one of the forms 8i ± 1: for N is of the form 2A2 − B2, where A is even and B odd, hence* any factor of it must be of the form2a2 − b2, that is, of the form 8i±1, and 2 must be a quadratic residue ofq. The theory of residues is, however, of but little use in finding factors of the cases that still await solution, though the possibility some day of finding a complete series of solutions by properties of residues must not be neglected†. Our present knowledge of the means of factorizingN has been summed up in the statement‡that a prime factor of the form2pt+1 can be found directly by rules due to Legendre, Gauss, and Jacobi, whent = 1, 3, 4, 8, or 12; and that a factor of the form2ptt′ + 1 can be found indirectly by a method due to Bickmore whent = 1, 3, 4, 8, or 12, and t′ is an odd integer greater than3. But this only indicates how little has yet been done towards finding a general solution of the problem. First. There is the simple but crude method of trying all possible prime divisors q which are of the form2pt + 1 as well as of one of the forms 8i ± 1. The chief known results for the smaller factors may be summarized by saying that a prime of this form will divideN when t = 1, ifp = 11, 23, 83, 131, 179, 191, 239, or 251; when t = 3, if p = 37, 73, or 233; when t = 5, if p = 43; when t = 15, if p = 113; when t = 17, if p = 79; when t = 19, if p = 29, or 197; when t = 25, if p = 47; when t = 41, if p = 223; when t = 59, if p = 97; when t = 163, if p = 41; when t = 1525, if p = 59; when t = 4, if p = 11, 29, 179, or 239; when t = 8, if p = 11; when t = 12, if p = 239; when t = 36, if p = 29, or211; when t = 60, if p = 53, or 151; and when t = 1445580, if p = 67. Of the 25 cases in which we know that Mersenne’s statement of the composite character ofN is correct all save 3 can be easily verified by trial in this way. For neglecting all values oft not exceeding, say, * Legendre, Théorie des Nombres, third edition, Paris, 1830, vol.i, § 143. In the case of Mersenne’s numbers,B = b = 1. † For methods of finding the residue indices of2see Bickmore,Messenger of Math- ematics, May, 1895, vol.xxv, pp. 15–21; see also Lieut-Colonel A.J.C. Cunning- ham on 2 as a 16-ic residue, Proceedings of the London Mathematical Society, 1895–6, vol.xxvii, pp. 85–122. ‡ Transactions of the British Association for the Advancement of Science(Ipswich Meeting), 1895, p. 614.

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[PDF page 247] CH. IX] MERSENNE’S NUMBERS. 231 60 which make q either composite or not of one of the forms8i ± 1 we have in each case only some20 or so divisors to try. Of the 3 other cases in which Mersenne’s statement of the composite character ofN has been verified, one verification (p = 41) is due to Plana, who frankly confessed that the result was reached “par un heureux hasard”; a second is due to Landry (p = 59), who did not explain how he obtained the factors; and the third is due to Cole (p = 67), who established it by the use of quadratic residues ofN, involving laborious numerical work. Of the 10 cases in which we know that Mersenne’s statement of the prime character of N is correct all save one may be verified by trial in this way, for the number of possible factors is not large. The exception is the case wherep equals 61, which Seelhoff and others have shown to be prime. Thus practically we may say that simple empirical trials would at once lead us to all the conclusions known except in the case ofp = 41 due to Plana, ofp = 59 due to Landry, ofp = 61 due to Seelhoff, and of p = 67 due to Cole. In fact, save for these four results the conclusions of all mathematicians to date could be obtained by anyone by a few hours’ arithmetical work. As p increases the number of factors to be tried increases so fast that, ifp is large, it would be practically impossible to apply the test to obtain large factors. This is an important point, for it has been asserted that in the cases still awaiting verification there are no factors less than 50, 000. Hence, we may take it as reasonably certain that this cannot have been the method by which the result was originally obtained; nor, as here enunciated, is it likely to give many factors not yet known. Of course it is possible there may be ways by which the number of possible values oft might be further limited, and if we could find them we might thus diminish the number of possible factors to be tried, but it will be observed that the values of N which still await verification are very large, for instance, whenp = 257, N contains no less than78 digits. It is hardly necessary to add that ifq is known and is not very large we can determine whether or not it is a factor ofN without the labour of performing the division. For instance, if we want to verify thatq = 13367 is a factor ofN when p = 41, we proceed thus. Take the power of2 nearest to q or to

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[PDF page 248] 232 MERSENNE’S NUMBERS. [CH. IX its square-root. We have to the modulusq 214 = 16384 ≡ 3017 ≡ 7 × 431 , ∴ 228 ≡ 49(−1377) ≡ −638 , ∴ 227 ≡ −319 , ∴ 214+27 ≡ (3017)(−319) ≡ 1 , ∴ 241 ≡ 1 . Second. We can proceed by reducing the problem to the solution of an indeterminate equation. It is clear that we can obtain a factor ofN if we can express it as the difference of the squares, or more generally of thenth powers, of two integers u and v. Further, if we can express a multiple ofN, say mN, in this form, we can find a factor ofmN and (with certain obvious limitations as to the value ofm) this will lead to a factor ofN. It may be also added that ifm can be found so thatN/m is expressible as a continued fraction of a certain form, a certain continuant* defined by the form of the continued fraction is a factor ofN. Since N can always be expressed as the difference of two squares, this method seems a natural one by which to attack the problem. If we put N = n2 + a = (n + b)2 − (b2 + 2bn − a), we can make use of the known forms ofu and v, and thus obtain an indeterminate equation between two variablesx and y of the form x2 = (2py + H)2 − 4(K − y) where H and K are numbers which can be easily calculated. Integral values of x and y where y < K will determine values ofu and v, and thus give factors of N. We can also attack the problem by indeterminate equations in an- * See J.G. Birch in theMessenger of Mathematics, August, 1902, vol.xxii, pp. 52– 55.

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[PDF page 249] CH. IX] MERSENNE’S NUMBERS. 233 other way. For the factors must be of the form2pt+1 and 8ps+1, hence (2pt + 1)(8ps + 1) = N , = 2p − 1 , = 2(2p−1 − 1) + 1, ∴ 4s + t + 8pst = (2p−1 − 1)/p , = (say)α + 8pβ . Hence 4s + t = α + 8px, and st = β − x , where x ≯ β and t is odd. These results again lead to an indetermi- nate equation. But, in both cases, unless p is small, the resulting equations are intractable. Third. A not uncommon method of attacking problems such as this, dealing with the factorization of large numbers, is through the theory of quadratic forms*. At best this is a difficult method to use, and we have no reason to think that it would have been employed by a mathematician of the seventeenth century. I here content myself with alluding to it. Fourth. There is yet another way in which the problem might be attacked. The problem will be solved if we can find an odd primeq so that to it as modulus2p+y ≡ z, and 2y ≡ z, where y and z may have any values we like to choose. If such values ofq, y, andz can be found, we have2y(2p − 1) ≡ 0. Therefore 2p = 1, that is,q is a divisor ofN. For example, to the modulus23, we have 28 ≡ 3 , 216 ≡ 32 . Also 25 ≡ 32 . Therefore 216 − 25 ≡ 0 , ∴ 211 − 1 ≡ 0 . Without going further into the matter we may say that theà priori determination of the values q, y, and z introduces us to an almost * For a sketch of this see G.B. Mathews,Theory of Numbers, part 1, Cambridge, 1891, pp. 261-271. See also F.N. Cole’s paper, On the Factoring of Large Num- bers, Bulletin of the American Mathematical Society, December, 1903, pp. 134- 137; andQuadratic Partitionsby A.J.C. Cunningham, London, 1904.

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[PDF page 250] 234 MERSENNE’S NUMBERS. [CH. IX untrodden field. It is just possible (though I should suppose unlikely) that the key to the riddle is to be found on methods of findingq, y, z, to satisfy the above conditions. For instance, if we could say what was the remainder when2x was divided by a primeq of the form2pt + 1, and if the remainders were the same whenx = u and x = v, then to the modulus q we should have,2u ≡ 2v, and therefore2u−v ≡ 1. It should however be noted that Jacobi’sCanon Arithmeticus and the similar canon drawn up by Cunningham would, if carried far enough, enable us to solve the problem by this method. Cunningham’s Canon gives the solution of the congruence2x ≡ R for all prime moduli less than1000, but it is of no use in determining factors ofN larger than 1000. It is however possible that ifR or q have certain forms such a canon might be constructed, and thus lead to a solution of the problem. Fifth. It is noteworthy that the odd values of p specified by Mersenne are primes of one of the four forms2q ± 1 or 2q ± 3, but it is not the case that all primes of these forms makeN a prime, for instance, N is composite ifp = 23 + 3 = 11 or if p = 25 − 3 = 29. This fact has suggested to more than one mathematician the possi- bility that some test as to the prime or composite character ofN when p is of one of these forms may be discoverable. Of course this is merely a conjecture. There is however this to say for it, that we know that Fermat* had paid attention to numbers of this form. Sixth. The numberN when expressed in the binary scale, consists of 1 repeated p times. This has suggested whether the work connected with the determination of factors ofN might not with advantage be expressed in the binary scale. A method based on the use of properties of this scale has been indicated by G. de Longchamps†, but as there given it would be unlikely to lead to the discovery of large divisors. I am, however, inclined to think that greater advantages would be gained by working in a scale whose radix was4p or may-be 8p—the resulting numbers being then expressed by a reasonably small number of digits. In fact when expressed in the latter scale in only one out of the 25 cases in which the factors ofN are known does the smallest factor contain more than two digits. * Ex. gr., see above, page 31. † Comptes rendus de l’Académie des Sciences, Paris, Nov. 1877, vol. lxxxv, pp. 950–952.

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[PDF page 251] CH. IX] MERSENNE’S NUMBERS. 235 Seventh. I have reserved to the last the description of the method which seems to me to be the most hopeful. We know by Fermat’s Theorem that ifx + 1is a prime then2x − 1 is divisible by x + 1. Hence if 2pt + 1 is a prime we have, to the modulus 2pt + 1 22pt − 1 ≡ 0 , ∴ (2p − 1)(1 + 2p + 22p + · · · + 2(2t−1)p) ≡ 0 . Hence, a divisor of2p − 1 will be known, if we can find a value oft such that 2pt + 1 is prime and the second factor is prime to it. This method may be used to establish Euler’s theorem of 1732. For if we putt = 1, and if2p + 1is a prime, we have, to the modulus2p + 1 (2p − 1)(2p + 1) ≡ 0 . Hence 2p ≡ 1 if 2p + 1is prime to2p + 1. This is the case ifp = 4m + 3. Hence 2p + 1 is a factor ofN if p = 11, 23, 83, 131, 179, 191, 239, and 251, for in these cases2p + 1 is prime. The problem of Mersenne’s Numbers is a particular case of the de- termination of the factors ofan −1. This has been the subject of inves- tigations by many mathematicians: an outline of their conclusions has been given by Bickmore*. I ought also to add a reference to the general method suggested by F.W. Lawrence†for the factorization of any high number: it is possible that Fermat used some method analogous to this. Finally, I should add that machines‡ have been devised for inves- tigating whether a number is prime, but I do not know that any have been constructed suitable for numbers as large as those involved in the numbers in question. * Messenger of Mathematics, Cambridge, 1895–6, vol.xxv, pp. 1–44; also 1896–7, vol. xxvi, pp. 1–38; see also a note by Mr E.B. Escott in theMessenger, 1903–4, vol. xxxiii, p. 49. † Ibid., 1894–5, vol.xxiv, pp. 100–109;Quarterly Journal of Mathematics, 1896, vol. xxviii, pp. 285–311; andTransactions of the London Mathematical Society, May 13, 1897, vol.xxviii, pp. 465–475. ‡ F.W. Lawrence, Quarterly Journal of Mathematics , 1896, already quoted, pp. 310–311; see also C.A. Laisant,Comptes Rendus Association Français pour l’avancement des sciences, 1891 (Marseilles), vol.xx, pp. 165–8.

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[PDF page 252] CHAPTER X. ASTROLOGY. Astrologers professed to be able to foretell the future, and within certain limits to control it. I propose to give in this chapter a concise account of the rules they used for this purpose†. I have not attempted to discuss the astrology of periods earlier than the middle ages, for the technical laws of the ancient astrology are not known with accuracy. At the same time there is no doubt that, as far back as we have any definite historical information, the art was practised in the East; that thence it was transplanted to Egypt, Greece, and Rome; and that the medieval astrology was founded on it. It is probable that the rules did not differ materially from those described in this chapter‡, and it may be added that the more intelligent thinkers of the old world recognised that the art had no valid pretences to accuracy. I may note also that the history of the development of the art ceases with the general acceptance of the Copernican theory, after which the practice of astrology rapidly became a mere cloak for imposture. † I have relied mainly on theManual of Astrologyby Raphael—whose real name was R.C. Smith—London, 1828, to which the references to Raphael hereafter given apply; and on Cardan’s writings, especially his commentary on Ptolemy’s work and hisGeniturarum Exempla. I am indebted also for various references and gossip to Whewell’sHistory of the Inductive Sciences; to various works by Raphael, published in London between 1825 and 1832; and to a pamphlet by M. Uhlemann, entitledGrundzüge der Astronomie und Astrologie, Leipzig, 1857. ‡ On the influences attributed to the planets, seeThe Dialogue of Bardesan on Fate, translated by W. Cureton in theSpicilegium Syriacum, London, 1855. 236

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[PDF page 253] CH. X] ASTROLOGY. 237 All the rules of the medieval astrology—to which I confine myself— are based on the Ptolemaic astronomy, and originate in theTetrabiblos* whichissaid, itmaybefalsely, tohavebeenwrittenbyPtolemyhimself. The art was developed by numerous subsequent writers, especially by Albohazen†, and Firmicus. The last of these collected the works of most of his predecessors in a volume‡, which remained a standard authority until the close of the sixteenth century. I may begin by reminding the reader that though there was a fairly general agreement as to the methods of procedure and interpretation— which alone I attempt to describe—yet there was no such thing as a fixed code of rules or a standard text-book. It is therefore difficult to reduce the rules to any precise and definite form, and almost impossi- ble, within the limits of a chapter, to give detailed references. At the same time the practice of the elements of the art was tolerably well established and uniform, and I feel no doubt that my account, as far as it goes, is substantially correct. There were two distinct problems with which astrologers concerned themselves. One was the determination in general outline of the life and fortunes of an enquirer: this was known asnatal astrology, and was effected by the erection of ascheme of nativity. The other was the means of answering any specific question about the individual: this was known ashorary astrology. Both depended on the casting or erecting of a horoscope. The person for whom it was erected was known as the native. A horoscope was cast according to the following rules§. The space between two concentric and similarly situated squares was divided into twelve spaces, as shown in the annexed diagram. These twelve spaces were known technically ashouses; they were numbered consecutively 1, 2, . . . ,12 (see figure); and were described as the first house, the se- cond house, and so on. The dividing lines were termedcusps: the line between the houses 12 and 1 was called the cusp of the first house, the line between the houses1 and 2 was called the cusp of the second house, and so on, finally the line between the houses11 and 12 was called the cusp of the twelfth house. Each house had also a name of its * There is an English translation by J. Wilson, London [n.d.]; and a French trans- lation is given in Halma’s edition of Ptolemy’s works. † De judiciis astrorum, ed. Liechtenstein, Basle, 1571. ‡ Astronomicorum, eight books, Venice, 1499. § Raphael, pp. 91–109.

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[PDF page 254] 238 ASTROLOGY. [CH. X 3 4 5 2 1 12 11 10 9 6 7 8 own—thus the first house was called the ascendant house, the eighth house was called the house of death, and so on—but as these names are immaterial for my purpose I shall not define them. Next, the positions which the various astrological signs and planets had occupied at some definite time and place (for instance, the time and place of birth of the native, if his nativity was being cast) were marked on the celestial sphere. This sphere was divided into twelve equal spaces by great circles drawn through the zenith, the angle be- tween any two consecutive circles being30◦. The first circle was drawn through the East point, and the space between it and the next circle towards the North corresponded to the first house, and sometimes was called the first house. The next space, proceeding from East to North, corresponded to the second house, and so on. Each of the twelve spaces between these circles corresponded to one of the twelve houses, and each of the circles to one of the cusps. In delineating* a horoscope, it was usual to begin by inserting the zodiacal signs. A zodiacal sign extends over30◦, and was marked on the cusp which passed through it: by its side was written a number indicating the distance to which its influence extended in the earlier of the two houses divided by the cusp. Next the position of the planets in these signs were calculated, and each planet was marked in its proper * Raphael, pp. 118–131.

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[PDF page 255] CH. X] ASTROLOGY. 239 house and near the cusp belonging to the zodiacal sign in which the planet was then situated: it was followed by a number indicating its right ascension measured from the beginning of the sign. The name of the native and the date for which the horoscope was cast were inserted usually in the central square. The diagram near the end of this chapter is a facsimile of the horoscope of Edward VI as cast by Cardan and will serve as an illustration of the above remarks. We are now in a position to explain how a horoscope wasread or interpreted. Each house was associated with certain definite questions and subjects, and the presence or absence in that house of the various signs and planets gave the answer to these questions or information on these subjects. These questions cover nearly every point on which information would be likely to be sought. They may be classified roughly as follows. For the answer, so far as it concerns the native, to all questions con- nected with his life and health, look in house 1; for questions connected with his wealth, refer to house 2; for his kindred and communications to him, refer to 3; for his parents and inheritances, refer to 4; for his children and amusements, refer to 5; for his servants and illnesses, refer to 6; for his marriage and amours, refer to 7; for his death, refer to 8; for his learning, religion and travels, refer to 9; for his trade and repu- tation, refer to 10; for his friends, refer to 11; and finally for questions connected with his enemies, refer to house 12. I proceed to describe briefly the influences of the planets, and shall then mention those of the zodiacal signs; I should note however that in practice the signs were in many respects more influential than the planets. The astrological “planets” were seven in number, and included the Sun and the Moon. They were Saturn or the Great Infortune, Jupiter or the Great Fortune, Mars or the Lesser Infortune, the Sun, Venus or the Lesser Fortune, Mercury, and the Moon: the above order being that of their apparent times of rotation round the earth. Each of them had a double signification. In the first place it im- pressed certain characteristics, such as good fortune, feebleness, &c., on the dealings of the native with the subjects connected with the house in which it was located; and in the second place it imported certain objects into the house which would affect the dealings of the native with the subjects of that house.

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[PDF page 256] 240 ASTROLOGY. [CH. X To describe the exact influence of each planet in each house would involve a long explanation, but the general effect of their presence may be indicated roughly as follows*. The presence of Saturn is malignant: that of Jupiter is propitious: that of Mars is on the whole injurious: that of the Sun indicates respectability and moderate success: that of Venus is rather favourable: that of Mercury implies rapid practical action: and lastly the presence of the Moon merely faintly reflects the influence of the planet nearest her, and suggests rapid changes and fickleness. Besides the planets, the Moon’s nodes and some of the more prominent fixed stars† also had certain influences. These vague terms may be illustrated by taking a few simple cases. For example, in casting a nativity, the life, health, and general career of the native were determined by the first or ascendant house, whence comes the expression that a man’s fortune is in the ascendant. NowthemostfavourableplanetwasJupiter. Therefore, ifattheinstant of birth Jupiter was in the first house, the native might expect a long, happy, healthy life; and being born under Jupiter he would have a “jovial” disposition. On the other hand, Saturn was the most unlucky of all the planets, and was as potent as malignant. If at the instant of birth he was in the first house, his potency might give the native a long life, but it would be associated with an angry and unhappy temper, a spirit covetous, revengeful, stern, and unloveable, though constant in friendship no less than in hate, which was what astrologers meant by a “saturnine” character. Similarly a native born under Mercury, that is, with Mercury in the first house, would be of a mercurial nature, while anyone born under Mars would have a martial bent. Moreover it was the prevalent opinion that a jovial person would have his horoscope affected by Jupiter, even if that planet had not been in the ascendant at the time of birth. Thus the horoscope of an adult depended to some extent on his character and previous life. It is hardly necessary to point out how easily this doctrine enabled an astrologer to make the prediction of the heavens agree with facts that were known or probable. In the same way the other houses are affected. For instance, no astrologer, who believed in the art, would have wished to start on a long journey when Saturn was in the ninth house or house of travels; * Raphael, pp. 70–90; pp. 204–209. † Raphael, pp. 129–131,
