Mathematical Recreations and Essays · W. W. Rouse Ball
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 of p which make a number of the form 2^p-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 form 2^p-1. In a memoir in the Messenger 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 Theorem Mersenne's enunciation of the results associated with his name is in the preface to his Cogitata (Cogitata Physico-Mathematica, Paris, 1644, praefatio generalis, article 19.). The passage is as follows:
"Vbi fuerit operae pretium aduertere xxviii numeros a Petro Bungo pro perfectis exhibitos, capite xxviii, libri de Numeris, non esse omnes Perfectos, quippe 20 sunt 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 ab 1 incipientis. Nonus enim perfectus est potestas exponentis 68 minus 1. Decimus, potestas exponentis
128 , minus 1. Vndecimus denique, potestas 258 , minus 1 , hoc est potestas 257 , vnitate decurtata, multiplicata per potestatem 256.
Qui vndecim alios repererit, nouerit se analysim omnem, quae fuerit hactenus, superasse: memineritque interea nullum esse perfectum à 17000 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 ad 2090000 , 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 difficultatibus, praescriptam numerorum perfectorum multitudinum exhibere; quemadmodum & agnoscere num dati numeri 15 , aut 20 caracteribus constantes, sint primi necne, cùm nequidem saeculum integrum huic examini, quocumque modo hactenus cognito, sufficiat.
It is evident that, if p is not a prime, then N is composite, and two or more of its factors can be written down by inspection. Hence we may confine ourselves to prime values of p. Mersenne, in effect, asserted that the only values of p , not greater than 257 , which make N a prime, are 1 , 2 , 3 , 5 , 7 , 13 , 17 , 19 , 31 , 67 , 127 , 257 : I assume that the number 67 is a misprint for 61. With this correction we have no reason to doubt the truth of the statement, but it has not been definitely established.
List of known results There are 56 primes not greater than 257. The determination of the prime or composite character of N for the 9 cases when p is less than 20 presents no difficulty: in only one of them is N composite. For 2 of the remaining 47 cases (namely, when p = 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 of N were given by Euler. He also proved that N was prime when p = 31. Plana gave the factors of N when p = 41. Landry and Le Lasseur discovered the factors in 10 cases (namely, when p=47 , 53 , 59 , 79 , 97 , 113 , 151 , 211 , 223 , and 233 ), but their analysis has not been published. Seelhoff showed that N was prime when p = 61 , Cunningham gave the factors when p = 197 , and Cole the factors when p = 67. Statements
have been made that the composite character of N when p = 89 , and its prime character when p=127 have been proved, but the proofs have not been published or verified.
Cases awaiting verification Thus there are 21 values of p for which Mersenne's statement still awaits verification. These are 71 , 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 when p= 127 , and 257 , and is composite for the other values, but as explained above it is probable that the character of N is known when p = 89 and 127.
To put the matter in another way. According to Mersenne's statement (corrected by the substitution of 61 for 67 ), 44 of the 56 primes less than 258 make N composite and the remaining 12 primes make N prime. In 25 out of the 44 cases in which N is said to be composite we know its factors, and in 19 cases the statement is still unverified. In 10 out of the 12 cases in which he said that N was prime his statement has been verified, and in 2 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 Fermat, 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 of N , and in the latter case its smallest factor, are given in the table table:Mersenne. 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 of N 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.
History of Investigations The factors of N when p=11 , 23 , and 37 had been indicated by Fermat (Oeuvres de Fermat, Paris, vol. ii, 1894, p. 210; or Opera Mathematica, Toulouse, 1679, p. 164; or Brassinne's Précis, Paris, 1853, p. 144.), some four years prior to the publication of Mersenne's work, in a letter dated October 18, 1640. The passage is as follows:
En la progression double, si d'un nombre quarré, généralement parlant, vous ôtez 2 ou 8 ou 32 &c., les nombres premiers moindres de l'unite qu'un multiple du quaternaire, qui mesureront le reste, feront l'effet requis. 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 reste 223 mesurera la 37e puissance -1 , &c.
The factors of N when p=29 , 43 , and 73 were given by Euler (Commentarii Academiae Scientiarum Petropolitanae, 1738, vol. vi, p. 103; or Commentationes Arithmeticae Collectae, vol. i, p. 2.) 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, if 4n + 3 and 8n + 7 are primes, then 2^(4n+3)-1 0 8n + 7. This was proved by Lagrange (Nouveaux Mémoires de l'Académie des Sciences de Berlin, 1775, pp. 323--356.) 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 (Histoire de l'Académie des Sciences for 1772, Berlin, 1774, p. 36. See also Legendre, Théorie des Nombres, third edition, Paris, 1830, vol. i, pp. 222--229.) in 1771. Fermat had asserted, in the letter mentioned
above, that the only possible prime factors of 2^p±1 , when p was prime, were of the form np + 1 , where n is an integer. This was proved by Euler (Novi Commentarii Academiae Scientiarum Petropolitanae, vol. i, p. 20; or Commentationes Arithmeticae Collectae, St Petersburg, 1849, vol. i, pp. 55, 56.) in 1748, who added that, since 2^p±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 define n much more closely, and Euler showed that the prime factors (if any) of 2^31-1 were necessarily primes of the form 248n + 1 or 248n + 63 ; also they must be less than sqrt(2^31-1) , that is, than 46339. Hence it is necessary to try only forty divisors to see if 2^31-1 is a prime or composite.
The factors of N when p = 41 were given by Plana (G.A.A. Plana, Memorie della Reale Accademia delle Scienze di Torino, Series 2, vol. xx, 1863, p. 130.) in 1859. He showed that the prime factors (if any) are primes of the form 328n + 1 or 328n + 247 , and lie between 1231 and sqrt(2^41-1) , that is, 1048573. Hence it is necessary to try only 513 divisors to see if 2^41-1 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 of p greater than 41. Plana (ibid:13Plana, p. 137.) added the forms of the prime divisors of N , if p is not greater than 101.
That N is prime when p = 127 seems to have been verified by Lucas (Sur la Théorie des Nombres Premiers, Turin, 1876, p. 11; and Recherches 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.) in 1876 and 1877. The demonstration has not been published.
The discovery of the factors of N for the values p = 47 , 53 , and 59 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 2^p-1 when p = 47 , 53 , and 59. He seems to have communicated his results to Lucas, who quoted them in the memoir cited below (American Journal of Mathematics, 1878, vol. i, p. 236.).
The factors of N when p = 79 and 113 were given first by Le Lasseur, and were quoted by Lucas in the same memoirLucas:footnote1:285Lucas:footnote2:285American Journal of Mathematics, 1878, vol. i, p. 236..
A factor of N when p = 233 was discovered later by Le Lasseur, and was quoted by Lucas in 1882 (Récréations, 1882--3, vol. i, p. 241.).
The factors of N when p = 97 , 151 , 211 , and 223 were determined subsequently by Le Lasseur, and were quoted by Lucas (ibid:14Récréations, vol. ii, p. 230.) in 1883.
That N is prime when p=61 had been conjectured by Landry and in 1886 a demonstration was offered by Seelhoff (P.H.H. Seelhoff, Zeitschrift für Mathematik und Physik, 1886, vol. xxxi, p. 178.). His demonstration is open to criticism, but the fact has been verified by others (See Weber-Wellstein, Encyclopaedie der Elementar-Mathematik, p. 48; and F.N. Cole, Bulletin of the American Mathematical Society, December, 1903, p. 136.), and may be accepted as proved.
That N is composite when p = 89 seems to have been verified by Lucas (Théorie des Nombres, Paris, 1891, p. 376.) in 1891, but the demonstration has not been published, nor have the actual factors been discovered.
That 7487 is a factor of N when p=197 was shown by A.J.C. Cunningham in 1895 (Proceedings of the London Mathematical Society, March 14, 1895, vol. xxvi, p. 261.).
That N is not prime when p = 67 seems to have been verified by Lucas (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, and Recherches sur les Ouvrages de Léonard de Pise, Rome, 1877, p. 26.) in 1876 and 1877. The composite nature ofCorrected: originally N 2^p=1 N=2^p - 1 when p = 67 was confirmed by E. Fauquembergue (L'Intermédiaire des mathématiciens, Paris, Sept. 1894, vol. i, p. 148.), and was also implied by Lucas (Théorie des Nombres, Paris, 1891, p. 376.) in 1891. The factors were given by F.N. Cole (On the Factoring of Large Numbers, Bulletin of the American Mathematical Society, December, 1903, pp. 134--137.) in 1903.
Bickmore in the memoir (C.E. Bickmore, Messenger of Mathematics, Cambridge, 1895, vol. xxv, p. 19.) cited below showed that 1433 is another factor of N if p = 179 ; and that 1913 and 5737 are other factors of N if p = 239.
Methods used in attacking the problem 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. The object is to find a prime divisor q (other than N and 1 ) of a number N when N is of the form 2^p-1 and p is a prime. It can be easily shown that q must be of the form 2pt + 1. Also q must be of one of the forms 8i ± 1 : for N is of the form 2A^2 - B^2 , where A is even and B odd, hence (Legendre, Théorie des Nombres, third edition, Paris, 1830, vol. i, § 143. In the case of Mersenne's numbers, B = b = 1 .) any factor of it must be of the form 2a^2 - b^2 , that is, of the form 8i ± 1 , and 2 must be a quadratic residue of q. 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 (For methods of finding the residue indices of 2 see Bickmore, Messenger of Mathematics, May, 1895, vol. xxv, pp. 15--21; see also Lieut-Colonel A.J.C. Cunningham on 2 as a 16 -ic residue, Proceedings of the London Mathematical Society, 1895--6, vol. xxvii, pp. 85--122.). Our present knowledge of the means of factorizing N has been summed up in the statement (Transactions of the British Association for the Advancement of Science (Ipswich Meeting), 1895, p. 614.) that a prime factor of the form 2pt + 1 can be found directly by rules due to Legendre, Gauss, and Jacobi, when t = 1 , 3 , 4 , 8 , or 12 ; and that a factor of the form 2ptt' + 1 can be found indirectly by a method due to Bickmore when t = 1 , 3 , 4 , 8 , or 12 , and t' is an odd integer greater than 3. But this only indicates how little has yet been done towards finding a general solution of the problem.
[By trial of divisors of known forms][Mersenne's Numbers] First There is the simple but crude method of trying all possible prime divisors q which are of the form 2pt + 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 divide N when t=1 , if p = 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 , or 211 ; 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 of N is correct all save 3 can be easily verified by trial in this way. For neglecting all values of t not exceeding, say, 60 which make q either composite or not of one of the forms 8i± 1 we have in each case only some 20 or so divisors to try. Of the 3 other cases in which Mersenne's statement of the composite character of
N 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 of N , 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 where p 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 of p = 41 due to Plana, of p = 59 due to Landry, of p = 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, if p 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 of t 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, when p = 257 , N contains no less than 78 digits.
It is hardly necessary to add that if q is known and is not very large we can determine whether or not it is a factor of N without the labour of performing the division.
For instance, if we want to verify that q = 13367 is a factor of N when p = 41 , we proceed thus. Take the power of 2 nearest to q or to its square-root. We have to the modulus q 0
[By indeterminate equations][Mersenne's Numbers] Second We can proceed by reducing the problem to the solution of an indeterminate equation.
It is clear that we can obtain a factor of N if we can express it as the difference of the squares, or more generally of the n th powers, of two integers u and v. Further, if we can express a multiple of N , say mN , in this form, we can find a factor of mN and (with certain obvious limitations as to the value of m ) this will lead to a factor of N. It may be also added that if m can be found so that N/m is expressible as a continued fraction of a certain form, a certain continuant (See J.G. Birch in the Messenger of Mathematics, August, 1902, vol. xxii, pp. 52--55.) defined by the form of the continued fraction is a factor of N.
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 = n^2 + a = (n + b)^2 - (b^2 + 2bn - a), we can make use of the known forms of u and v , and thus obtain an indeterminate equation between two variables x and y of the form x^2 = (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 of u and v , and thus give factors of N.
We can also attack the problem by indeterminate equations in another way. For the factors must be of the form 2pt + 1 and 8ps + 1 , hence (2pt + 1) (8ps + 1) = N,; = 2^p - 1,; = 2(2^(p - 1) - 1) + 1,; Therefore 4s + t + 8pst = (2^(p - 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 indeterminate equation.
But, in both cases, unless p is small, the resulting equations are intractable.
[By properties of quadratic forms][Mersenne's Numbers] 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 (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 Numbers, Bulletin of the American Mathematical Society, December, 1903, pp. 134-137; and Quadratic Partitions by A.J.C. Cunningham, London, 1904.). 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.
[By the use of a Canon Arithmeticus][Mersenne's Numbers] Fourth There is yet another way in which the problem might be attacked. The problem will be solved if we can find an odd prime q so that to it as modulus 2^(p+y) z , and 2^y z , where y and z may have any values we like to choose. If such values of q , y , and z can be found, we have 2^y (2^p - 1) 0. Therefore 2^p = 1 , that is, q is a divisor of N.
For example, to the modulus 23 , we have 2^8 3,; 2^16 3^2.; Also 2^5 3^2.; Therefore 2^16 - 2^5 0,; Therefore 2^11 - 1 0.; Without going further into the matter we may say that the à priori determination of the values q , y , and z introduces us to an almost untrodden field. It is just possible (though I should suppose unlikely) that the key to the riddle is to be found on methods of finding q , y , z , to satisfy the above conditions. For instance, if we could say what was the remainder when 2^x was divided by a prime q of the form 2pt + 1 , and if the remainders were the same when x = u and x = v , then to the modulus q we should have, 2^u 2^v , and therefore 2^(u-v) 1.
It should however be noted that Jacobi's Canon 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 congruence 2^x R for all prime moduli less than 1000 , but it is of no use in determining factors of N larger than 1000. It is however possible that if R or q have certain forms such a canon might be constructed, and thus lead to a solution of the problem.
[By properties of binary powers][Mersenne's Numbers] Fifth It is noteworthy that the odd values of p specified by Mersenne are primes of one of the four forms 2^q ± 1 or 2^q ± 3 , but it is not the case that all primes of these forms make N a prime, for instance, N is composite if p = 2^3 + 3 = 11 or if p = 2^5 - 3 = 29.
This fact has suggested to more than one mathematician the possibility that some test as to the prime or composite character of N 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 (, see above, page .) had paid attention to numbers of this form.
[By the use of the binary scale][Mersenne's Numbers] Sixth The number N when expressed in the binary scale, consists of 1 repeated p times. This has suggested whether the work connected with the determination of factors of N 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 (Comptes rendus de l'Académie des Sciences, Paris, Nov. 1877, vol. lxxxv, pp. 950--952.), 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 was 4p 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 of N are known does the smallest factor contain more than two digits.
[By the use of Fermat's Theorem][Mersenne's Numbers] 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 if x + 1 is a prime then 2^x-1 is divisible by x + 1. Hence if 2pt + 1 is a prime we have, to the modulus 2pt + 1 2^(2pt)-1 0,; Therefore (2^p-1)(1+2^p+2^2p+...+2^((2t-1)p)) 0. Hence, a divisor of 2^p-1 will be known, if we can find a value of t 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 put t=1 , and if 2p+1 is a prime, we have, to the modulus 2p + 1 (2^p-1)(2^p + 1) 0. Hence 2^p1 if 2^p + 1 is prime to 2p+1. This is the
case if p = 4m + 3. Hence 2p + 1 is a factor of N if p = 11 , 23 , 83 , 131 , 179 , 191 , 239 , and 251 , for in these cases 2p + 1 is prime.
The problem of Mersenne's Numbers is a particular case of the determination of the factors of a^n - 1. This has been the subject of investigations by many mathematicians: an outline of their conclusions has been given by Bickmore (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 the Messenger, 1903--4, vol. xxxiii, p. 49.). I ought also to add a reference to the general method suggested by F.W. Lawrence (ibid:15Messenger of Mathematics, Cambridge, 1894--5, vol. xxiv, pp. 100--109; Quarterly Journal of Mathematics, 1896, vol. xxviii, pp. 285--311; and Transactions of the London Mathematical Society, May 13, 1897, vol. xxviii, pp. 465--475.) for the factorization of any high number: it is possible that Fermat used some method analogous to this.
Mechanical methods of Factorizing Numbers Finally, I should add that machines (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.) have been devised for investigating 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 questionThe primality or otherwise of all Mersenne numbers 2^p-1 up to p=6972593 has been decided: thirty-eight are prime (see http://www.mersenne.orgwww.mersenne.org). Of the asterisked entries in the author's table on page , 2^89-1 , 2^(107)-1 and 2^(127)-1 are prime and the rest (including 2^(257)-1) are composite..
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