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[PDF page 233] CH. VIII] APPROXIMATIONS TO THE V ALUE OF π. 217 Vieta*, in 1579, showed thatπ was greater than31415926535/1010, and less than31415926537/1010. This was deduced from the perimeters of the inscribed and circumscribed polygons of6 × 216 sides, obtained by repeated use of the formula2 sin2 1 2 θ = 1 − cos θ. He also gave† a result equivalent to the formula 2 π = √2 2 √(2 + √2) 2 √{2 + √(2 + √2)} 2 · · · . The father of Adrian Metius‡, in 1585, gave355/113, which is equal to 3.14159292 . . ., and is correct to6 places of decimals. This was a cu- riousandluckyguess, forallthatheprovedwasthat π wasintermediate between 377/120 and 333/106, whereon he jumped to the conclusion that he should obtain the true fractional value by taking the mean of the numerators and the mean of the denominators of these fractions. In 1593 Adrian Romanus§calculated the perimeter of the inscribed regular polygon of 1073, 741824 (i.e. 230) sides, from which he deter- mined the value ofπ correct to 15 places of decimals. L. van Ceulen devoted no inconsiderable part of his life to the subject. In 1596∥ he gave the result to20 places of decimals: this was calculated by finding the perimeters of the inscribed and circumscribed regular polygons of 60 × 233 sides, obtained by the repeated use of a theorem of his discovery equivalent to the formula1−cos A = 2 sin2 1 2 A. I possess a finely executed engraving of him of this date, with the result printed round a circle which is below his portrait. He died in 1610, and by his directions the result to35 places of decimals (which was as far as he had then calculated it) was engraved on his tombstone¶ in St Peter’s Church, Leyden. His posthumous arithmetic** contains the result to32 places; this was obtained by calculating the perimeter of a polygon, the * Canon Mathematicus seu ad Triangula, Paris, 1579, pp. 56, 66: probably this work was printed for private circulation only, it is very rare. † Vietae Opera, ed. Schooten, Leyden, 1646, p. 400. ‡ Arithmeticae libri duo et Geometriae, by A. Metius, Leyden, 1626, pp. 88, 89. [Probably issued originally in 1611.] § Ideae Mathematicae, Antwerp, 1593: a rare work, which I have never been able to consult. ∥ Vanden Circkel, Delf, 1596, fol. 14, p. 1; orDe Circulo, Leyden, 1619, p. 3. ¶ The inscription is quoted by Prof. de Haan in theMessenger of Mathematics, 1874, vol. iii, p. 25. ** De Arithmetische en Geometrische Fondamenten, Leyden, 1615, p.163; orp.144 of the Latin translation by W. Snell, published at Leyden in 1615 under the

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[PDF page 234] 218 THREE GEOMETRICAL PROBLEMS. [CH. VIII number of whose sides is262, i.e. 4, 611686, 018427, 387904. Van Ceulen also compiled a table of the perimeters of various regular polygons. Willebrord Snell*, in 1621, obtained from a polygon of230 sides an approximation to 34 places of decimals. This is less than the numbers given by van Ceulen, but Snell’s method was so superior that he ob- tained his34 places by the use of a polygon from which van Ceulen had obtained only14 (or perhaps16) places. Similarly, Snell obtained from a hexagon an approximation as correct as that for which Archimedes had required a polygon of96 sides, while from a polygon of96 sides he determined the value ofπ correct to seven decimal places instead of the two places obtained by Archimedes. The reason is that Archimedes, havingcalculatedthelengthsofthesidesofinscribedandcircumscribed regular polygons of n sides, assumed that the length of1/nth of the perimeter of the circle was intermediate between them; whereas Snell constructed from the sides of these polygons two other lines which gave closer limits for the corresponding arc. His method depends on the the- orem 3 sinθ/(2 + cos θ) < θ < (2 sin 1 3 θ + tan 1 3 θ), by the aid of which a polygon of n sides gives a value ofπ correct to at least the integral part of (4 logn − .2305) places of decimals, which is more than twice the number given by the older rule. Snell’s proof of his theorem is incorrect, though the result is true. Snell also added a table†of the perimeters of all regular inscribed and circumscribed polygons, the number of whose sides is10×2n where n is not greater than 19 and not less than 3. Most of these were quoted from van Ceulen, but some were recalculated. This list has proved useful in refuting circle-squarers. A similar list was given by James Gregory‡. In 1630 Grienberger§, by the aid of Snell’s theorem, carried the approximation to39 places of decimals. He was the last mathematician who adopted the classical method of finding the perimeters of inscribed and circumscribed polygons. Closer approximations serve no useful title Fundamenta Arithmetica et Geometrica. This was reissued, together with a Latin translation of theVanden Circkel, in 1619, under the titleDe Circulo; in which see pp. 3, 29–32, 92. * Cyclometricus, Leyden, 1621, p. 55. † It is quoted by Montucla, ed. 1831, p. 70. ‡ Vera Circuli et Hyperbolae Quadratura, prop. 29, quoted by Huygens, Opera Varia, Leyden, 1724, p. 447. § Elementa Trigonometrica, Rome, 1630, end of preface.

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[PDF page 235] CH. VIII] APPROXIMATIONS TO THE V ALUE OF π. 219 purpose. Proofs of the theorems used by Snell and other calculators in applying this method were given by Huygens in a work* which may be taken as closing the history of this method. In 1656 Wallis† proved that π 2 = 2 · 2 · 4 · 4 · 6 · 6 · · · 1 · 3 · 3 · 5 · 5 · 7 · 7 · · · , and quoted a proposition given a few years earlier by Viscount Brouncker to the effect that π 4 = 1 + 12 2 + 32 2 + 52 2 + . . . , but neither of these theorems was used to any large extent for cal- culation. Subsequent calculators have relied on converging infinite series, a method that was hardly practicable prior to the invention of the calcu- lus, though Descartes‡ had indicated a geometrical process which was equivalent to the use of such a series. The employment of infinite series was proposed by James Gregory§, who established the theorem that θ = tan θ − 1 3 tan3 θ + 1 5 tan5 θ − · · ·, the result being true only ifθ lies between − 1 4 π and 1 4 π. The first mathematician to make use of Gregory’s series for ob- taining an approximation to the value ofπ was Abraham Sharp∥, who, in 1699, on the suggestion of Halley, determined it to 72 places of * De Circula Magnitudine Inventa, 1654; Opera Varia, pp. 351–387. The proofs are given in G. Pirie’sGeometrical Methods of Approximating the Value ofπ, London, 1877, pp. 21–23. † Arithmetica Infinitorum, Oxford, 1656, prop. 191. An analysis of the investi- gation by Wallis was given by Cayley,Quarterly Journal of mathematics, 1889, vol. xxiii, pp. 165–169. ‡ See Euler’s paper in theNovi Commentarii Academiae Scientiarum, St Peters- burg, 1763, vol.viii, pp. 157–168. § See the letter to Collins, dated Feb. 15, 1671, printed in theCommercium Epis- tolicum, London, 1712, p. 25, and in the Macclesfield Collection,Correspondence of Scientific Men of the Seventeenth Century, Oxford, 1841, vol.ii, p. 216. ∥ See Life of A. Sharpby W. Cudworth, London, 1889, p. 170. Sharp’s work is given in one of the preliminary discourses (p. 53et seq.) prefixed to H. Sherwin’s Mathematical Tables. The tables were issued at London in 1705: probably the discourses were issued at the same time, though the earliest copies I have seen were printed in 1717.

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[PDF page 236] 220 THREE GEOMETRICAL PROBLEMS. [CH. VIII decimals (71 correct). He obtained this value by putting θ = 1 6 π in Gregory’s series. Machin*, earlier than 1706, gave the result to100 places (all cor- rect). He calculated it by the formula 1 4 π = 4 tan−1 1 5 − tan−1 1 239 . De Lagny†, in 1719, gave the result to127 places of decimals (112 correct), calculating it by puttingθ = 1 6 π in Gregory’s series. Hutton‡, in 1776, and Euler§, in 1779, suggested the use of the formulae 1 4 π = tan −1 1 2 + tan−1 1 3 or 1 4 π = 5 tan−1 1 7 + 2 tan−1 3 79, but neither carried the approximation as far as had been done previously. Vega, in 1789∥, gave the value ofπ to 143 places of decimals (126 correct); and, in 1794¶, to 140 places (136 correct). Towards the end of the last century Baron Zach saw in the Radcliffe Library, Oxford, a manuscript by an unknown author which gives the value of π to 154 places of decimals (152 correct). In 1841 Rutherford** calculated it to 208 places of decimals (152 correct), using the formula1 4 π = 4 tan−1 1 5 − tan−1 1 70 + tan−1 1 99. In 1844 Dase††calculated it to205 places of decimals (200 correct), using the formula 1 4 π = tan −1 1 2 + tan−1 1 5 + tan−1 1 8. In 1847 Clausen ‡‡ carried the approximation to 250 places of decimals (248 correct), calculating it independently by the formulae 1 4 π = 2 tan−1 1 3 + tan−1 1 7 and 1 4 π = 4 tan−1 1 5 − tan−1 1 239. In1853Rutherford §§carriedhisformerapproximationto 440places of decimals (all correct), and William Shanks prolonged the approxima- tionto 530places. InthesameyearShankspublishedanapproximation * W. Jones’sSynopsis Palmariorum, London, 1706, p. 243; and Maseres,Scrip- tores Logarithmici, London, 1796, vol.iii, pp. vii–ix, 155–164. † Histoire de l’Académiefor 1719, Paris, 1721, p. 144. ‡ Philosophical Transactions, 1776, vol.lxvi, pp. 476–492. § Nova Acta Academiae Scientiarum Petropolitanaefor 1793, St Petersburg, 1798, vol. xi, pp. 133–149: the memoir was read in 1779. ∥ Nova Acta Academiae Scientiarum Petropolitanaefor 1790, St Petersburg, 1795, vol. ix, p. 41. ¶ Thesaurus Logarithmorum (logarithmisch-trigonometrischer Tafeln), Leipzig, 1794, p. 633. ** Philosophical Transactions, 1841, p. 283. ††Crelle’s Journal, 1844, vol.xxvii, p. 198. ‡‡Schumacher,Astronomische Nachrichten, vol. xxv, col. 207. §§ Proceedings of the Royal Society, Jan. 20, 1853, vol.vi, pp. 273-275.

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[PDF page 237] CH. VIII] APPROXIMATIONS TO THE V ALUE OF π. 221 to 607 places*: and in 1873 he carried the approximation to707 places of decimals†. These were calculated from Machin’s formula. In 1853 Richter, presumably in ignorance of what had been done in England, foundthevalueof π to 333places‡ofdecimals( 330correct); in 1854 he carried the approximation to400 places§; and in 1855 carried it to 500 places∥. Of the series and formulae by which these approximations have been calculated, those used by Machin and Dase are perhaps the easiest to employ. Other series which converge rapidly are the following, π 6 = 1 2 + 1 2 · 1 3 · 23 + 1 · 3 2 · 4 · 1 5 · 25 + · · · , and π 4 = 2 + 22 tan−1 1 28 + tan−1 1 443 − 5 tan−1 1 1393 − 10 tan−1 1 11018 , the latter of these is due to Mr Escott¶. As to those writers who believe that they have squared the circle their number is legion and, in most cases, their ignorance profound, but their attempts are not worth discussing here. “Only prove to me that it is impossible,” said one of them, “and I will set about it immediately”; and doubtless the statement that the problem is insoluble has attracted much attention to it. Among the geometrical ways of approximating to the truth the following is one of the simplest. Inscribe in the given circle a square, and to three times the diameter of the circle add a fifth of a side of the square, the result will differ from the circumference of the circle by less than one-seventeen-thousandth part of it. An approximate value ofπ has been obtained experimentally by the theory of probability. On a plane a number of equidistant parallel straight lines, distance aparta, are ruled; and a stick of lengthl, which * Contributions to Mathematics, W. Shanks, London, 1853, pp. 86, 87. † Proceedings of the Royal Society, 1872–3, vol.xxi, p. 318; 1873–4, vol.xxii, p. 45. ‡ Grunert’s Archiv, vol. xxi, p. 119. § Ibid., vol. xxiii, p. 476: the approximation given in vol.xxii, p. 473, is correct only to 330 places. ∥ Ibid., vol. xxv, p. 472; andElbingen Anzeigen, No. 85. ¶ L’Intermédiaire des mathématiciens, Paris, Dec. 1896, vol.iii, p. 276.

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[PDF page 238] 222 THREE GEOMETRICAL PROBLEMS. [CH. VIII is less thana, is dropped on to the plane. The probability that it will fall so as to lie across one of the lines is2l/πa. If the experiment is repeated many hundreds of times, the ratio of the number of favourable cases to the whole number of experiments will be very nearly equal to this fraction: hence the value ofπ can be found. In 1855 Mr A. Smith* of Aberdeen made3204trials, and deducedπ = 3.1553. A pupil of Prof. De Morgan*, from600 trials, deducedπ = 3.137. In 1864 Captain Fox† made 1120 trials with some additional precautions, and obtained as the mean value π = 3 .1419. Other similar methods of approximating to the value ofπ have been indicated. For instance, it is known that if two numbers are written down at random, the probability that they will be prime to each other is 6/π2. Thus, in one case‡ where each of 50 students wrote down 5 pairs of numbers at random,154 of the pairs were found to consist of numbers prime to each other. This gives6/π2 = 154/250, from which we get π = 3 .12. * A. De Morgan, Budget of Paradoxes, London, 1872, pp. 171, 172 [quoted from an article by De Morgan published in 1861]. † Messenger of Mathematics, Cambridge, 1873, vol.ii, pp. 113, 114. ‡ Note onπ by R. Chartres.Philosophical Magazine, London, series 6, vol.xxxix, March, 1904, p. 315.

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[PDF page 239] CHAPTER IX. MERSENNE’S NUMBERS. One of the unsolved riddles of higher arithmetic, to which I have alluded in Chapter I, is the discovery of the method by which Mersenne or his contemporaries determined values ofp which make a number of the form 2p − 1 a prime. It is convenient to describe such primes as Mersenne’s Numbers. In this chapter, for shortness, I use N to denote a number of the form2p − 1. In a memoir in theMessenger of Mathematics in 1891 I gave a brief sketch of the history of the problem. I here repeat the facts in somewhat more detail, and add a sketch of methods used in attacking the problem. Mersenne’s enunciation of the results associated with his name is in the preface to hisCogitata†. The passage is as follows: “Vbi fuerit operae pretium aduerterexxviii numeros a Petro Bungo pro perfectis exhibitos, capite xxviii, libri de Numeris, non esse omnes Perfectos, quippe20sunt imperfecti, adeovt [adeunt?] solos octo perfectos habeat...qui sunt è regione tabulae Bungi, 1, 2, 3, 4, 8, 10, 12, et 29: quique soli perfecti sunt, vt qui Bungum habuerint, errori medicinam faciant. Porrò numeri perfecti adeo rari sunt, vt vndecim dumtaxat potuerint hactenus inueniri: hoc est, alii tres a Bongianis differentes: neque enim vllus est alius perfectus ab illis octo, nisi superes exponentem numerum 62, progressionis duplae ab1 incipientis. Nonus enim perfectus est potes- tas exponentis 68 minus 1. Decimus, potestas exponentis 128, minus 1. Vndecimus denique, potestas258, minus 1, hoc est potestas257, vnitate decurtata, multiplicata per potestatem256. Qui vndecim alios repererit, nouerit se analysim omnem, quae fuerit hactenus, superasse: memineritque interea nullum esse perfectum à17000 † Cogitata Physico-Mathematica, Paris, 1644, praefatio generalis, article 19. 223

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[PDF page 240] 224 MERSENNE’S NUMBERS. [CH. IX potestate ad 32000; & nullum potestatum interuallum tantum assignari posse, quin detur illud absque perfectis. Verbi gratia, si fuerit exponens 1050000, nullus erit numerus progressionis duplae vsque ad2090000, qui perfectis numeris seruiat, hoc est qui minor vnitate, primus existat. Vnde clarum est quàm rari sint perfecti numeri, & quàm merito viris perfectis comparentur; esseque vnam ex maximis totius Matheseos diffi- cultatibus, praescriptam numerorum perfectorum multitudinum exhibere; quemadmodum & agnoscere num dati numeri15, aut20 caracteribus con- stantes, sint primi necne, cùm nequidem saeculum integrum huic examini, quocumque modo hactenus cognito, sufficiat. It is evident that, ifp is not a prime, thenN is composite, and two or more of its factors can be written down by inspection. Hence we may confine ourselves to prime values ofp. Mersenne, in effect, asserted that the only values ofp, not greater than257, which makeN a prime, are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257: I assume that the number67 is a misprint for61. With this correction we have no reason to doubt the truth of the statement, but it has not been definitely established. There are 56 primes not greater than257. The determination of the prime or composite character ofN for the 9 cases whenp is less than 20 presents no difficulty: in only one of them is N composite. For 2 of the remaining 47 cases (namely, whenp = 23 and 37) the decomposition of N had been given by Fermat. For 9 of them (namely, when p = 29, 43, 73, 83, 131, 179, 191, 239, 251) the factors ofN were given by Euler. He also proved thatN was prime whenp = 31. Plana gave the factors ofN when p = 41. Landry and Le Lasseur discovered the factors in 10 cases (namely, whenp = 47, 53, 59, 79, 97, 113, 151, 211, 223, and 233), but their analysis has not been published. Seelhoff showed that N was prime whenp = 61, Cunningham gave the factors when p = 197, and Cole the factors whenp = 67. Statements have been made that the composite character ofN when p = 89, and its prime character whenp = 127 have been proved, but the proofs have not been published or verified. Thus there are21 values of p for which Mersenne’s statement still awaits verification. These are71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, 257. For these values N is (according to Mersenne) prime whenp = 127, and257, and is composite for the other values, but as explained above it is probable that the character ofN is known whenp = 89 and 127. To put the matter in another way. According to Mersenne’s state- ment (corrected by the substitution of61 for 67), 44 of the 56 primes

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[PDF page 241] CH. IX] MERSENNE’S NUMBERS. 225 less than258 make N composite and the remaining12 primes makeN prime. In 25 out of the44 cases in whichN is said to be composite we know its factors, and in19 cases the statement is still unverified. In10 out of the 12 cases in which he said thatN was prime his statement has been verified, and in2 cases it is still unverified. From the wording of the last clause in the above quotation it has been conjectured that the result had been communicated to Mersenne, and that he published it without being aware of how it was proved. In itself this seems probable. He was a good mathematician, but not an exceptional genius. It would be strange if he established a proposition which has baffled Euler, Lagrange, Legendre, Gauss, Jacobi, and other mathematicians of the first rank; but if the proposition is due to Fer- mat, with whom Mersenne was in constant correspondence, the case is altered, and not only is the absence of a demonstration explained, but we cannot be sure that we have attacked the problem on the best lines. The known results as to the prime or composite character ofN, and in the latter case its smallest factor, are given in the table on the following page. The cases that remain as yet unverified are marked with an asterisk. Before describing the methods used for attacking the problem it will be convenient to state in more detail when and by whom these results were established. The factors (if any) of such values ofN as are less than a million can be verified easily: they have been known for a long time, and I need not allude to them in detail. The factors of N when p = 11, 23, and 37 had been indicated by Fermat*, some four years prior to the publication of Mersenne’s work, in a letter dated October 18, 1640. The passage is as follows: Enlaprogressiondouble, sid’unnombrequarré, généralementparlant, vous ôtez 2 ou 8 ou 32 &c., les nombres premiers moindres de l’unite qu’unmultipleduquaternaire, quimesurerontlereste, ferontl’effetrequis. Comme de 25, qui est un quarré, ôtez 2; le reste 23 mesurera la 11e puissance −1; ôtez 2 de 49, le reste 47 mesurera la 23e puissance −1. Ôtez 2 de 225, le reste223 mesurera la 37e puissance −1, &c. The factors of N when p = 29, 43, and 73 were given by Euler† * Oeuvres de Fermat, Paris, vol.ii, 1894, p. 210; orOpera Mathematica, Toulouse, 1679, p. 164; or Brassinne’sPrécis, Paris, 1853, p. 144. † Commentarii Academiae Scientiarum Petropolitanae, 1738, vol. vi, p. 103; or Commentationes Arithmeticae Collectae, vol. i, p. 2.

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[PDF page 242] 226 MERSENNE’S NUMBERS. [CH. IX p Value of N = 2 p − 1 1 1 prime 2 3 prime 3 7 prime 5 31 prime 7 127 prime 11 2047 = 23 × 89 composite 13 8191 prime 17 131071 prime 19 524287 prime 23 8388607 = 47 × 178481 composite Fermat 29 536870911 = 233 × 1103 × 2089 composite Euler 31 2147483647 prime Euler 37 137438953471 = 223 × 616318177 composite Fermat 41 2199023255551 = 13367 × 164511353 composite Plana 43 8796093022207 = 431 × 9719 × 2099863 composite Euler 47 2351 × 4513 × 13264529 composite Landry 53 6361 × 69431 × 20394401 composite Landry 59 179951 × 3203431780337 composite Landry 61 2305843009213693951 prime Seelhoff 67 ≡ 0 (193707721) composite Cole 71 2361183241434822606847 ∗ 73 ≡ 0 (439) composite Euler 79 ≡ 0 (2687) composite Le Lasseur 83 ≡ 0 (167) composite Euler 89 618970019642690137449562111 ∗ 97 ≡ 0 (11447) composite Le Lasseur 101 2535301200456458802993406410751 ∗ 103 10141204801825835211973625643007 ∗ 107 162259276829213363391578010288127 ∗ 109 649037107316853453566312041152511 ∗ 113 ≡ 0 (3391) composite Le Lasseur 127 170141183460469231731687303715884105727 ∗ 131 ≡ 0 (263) composite Euler 137 .................................... ∗ 139 .................................... ∗ 149 .................................... ∗ 151 ≡ 0 (18121) composite Le Lasseur 157 .................................... ∗ 163 .................................... ∗ 167 .................................... ∗

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[PDF page 243] CH. IX] MERSENNE’S NUMBERS. 227 p Value of N = 2 p − 1 173 .................................... ∗ 179 ≡ 0 (359) composite Euler 181 .................................... ∗ 191 ≡ 0 (383) composite Euler 193 .................................... ∗ 197 ≡ 0 (7487) composite Cunningham 199 .................................... ∗ 211 ≡ 0 (15193) composite Le Lasseur 223 ≡ 0 (18287) composite Le Lasseur 227 .................................... ∗ 229 .................................... ∗ 233 ≡ 0 (1399) composite Le Lasseur 239 ≡ 0 (479) composite Euler 241 .................................... ∗ 251 ≡ 0 (503) composite Euler 257 .................................... ∗ in 1732. The fact that N is composite for the values p = 83, 131, 179, 191, 239, and 251 follows from a proposition enunciated, at the same time, by Euler to the effect that, if4n + 3and 8n + 7are primes, then 24n+3 − 1 ≡ 0 (mod 8 n + 7). This was proved by Lagrange* in his classical memoir of 1775. The proposition also covers the cases of p = 11 and p = 23. This is the only general theorem on the subject which is yet established. The fact that N is prime when p = 31 was proved by Euler† in 1771. Fermat had asserted, in the letter mentioned above, that the only possible prime factors of2p ± 1, when p was prime, were of the form np + 1, where n is an integer. This was proved by Euler‡in 1748, who added that, since 2p ± 1 is odd, every factor of it must be odd, and therefore if p is odd n must be even. But if p is a given number we can definen much more closely, and Euler showed that the prime factors (if any) of231 − 1 were necessarily primes of the form248n + 1 or 248n+63; also they must be less than √ 231 − 1, that is, than46339. * Nouveaux Mémoires de l’Académie des Sciences de Berlin, 1775, pp. 323–356. † Histoire de l’Académie des Sciencesfor 1772, Berlin, 1774, p. 36. See also Legendre, Théorie des Nombres, third edition, Paris, 1830, vol.i, pp. 222–229. ‡ Novi Commentarii Academiae Scientiarum Petropolitanae, vol.i, p. 20; orCom- mentationes Arithmeticae Collectae, St Petersburg, 1849, vol.i, pp. 55, 56.

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[PDF page 244] 228 MERSENNE’S NUMBERS. [CH. IX Hence it is necessary to try only forty divisors to see if231 − 1 is a prime or composite. The factors of N when p = 41 were given by Plana* in 1859. He showed that the prime factors (if any) are primes of the form328n+1 or 328n+247, and lie between1231 and √ 241 − 1, that is,1048573. Hence it is necessary to try only513 divisors to see if241 − 1 is composite: the seventeenth of these divisors gives the required factors. This is the same method of attacking the problem which was used by Euler in 1771, but it would be very laborious to employ it for values ofp greater than 41. Plana † added the forms of the prime divisors ofN, if p is not greater than 101. That N is prime when p = 127 seems to have been verified by Lucas‡in 1876 and 1877. The demonstration has not been published. The discovery of the factors ofN for the valuesp = 47, 53, and59 is due apparently to the late F. Landry, who established theorems on the factors (if any) of numbers of certain forms. Instead of publishing his results he issued a challenge to all mathematicians to solve the general problem. This is contained in a rare pamphlet published at Paris in 1867, of which I possess a copy, in which the factors of certain numbers are given, and on page 8 of which it is implied that he had obtained the factors of 2p − 1 when p = 47, 53, and 59. He seems to have communicated his results to Lucas, who quoted them in the memoir cited below§. The factors of N when p = 79 and 113 were given first by Le Lasseur, and were quoted by Lucas in the same memoir§. A factor of N when p = 233 was discovered later by Le Lasseur, and was quoted by Lucas in 1882∥. The factors ofN when p = 97, 151, 211, and 223 were determined subsequently by Le Lasseur, and were quoted by Lucas¶ in 1883. * G.A.A. Plana, Memorie della Reale Accademia delle Scienze di Torino, Series 2, vol. xx, 1863, p. 130. † Ibid., p. 137. ‡ Sur la Théorie des Nombres Premiers, Turin, 1876, p. 11; andRecherches sur les Ouvrages de Léonard de Pise, Rome, 1877, p. 26, quoted by Lieut.-Colonel A.J.C. Cunningham,Proceedings of the London Mathematical Society, Nov. 14, 1895, vol. xxvii, p. 54. § American Journal of Mathematics, 1878, vol.i, p. 236. ∥ Récréations, 1882–3, vol.i, p. 241. ¶ Ibid., vol. ii, p. 230.
